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LECTURES ON THE IKOSAHEDROK WMwm To illustrate page 2(i, f< acr;,. Tlie stereographio projection of the sphere on the plane is made from the point Z=0. r - S''TS'TK-T LECTUKES THE IKOSAHEDRON, AND THE SOLUTION OF EQUATIONS OF THE FIFTH DEGREE. FELIX KLEIN, PROFESSOR OF MATHEMATICS, GOTTINGEN. TRANSLATED BY GEORGE GAVIN MORRICE, M.A., M.B. MEMBER OF THE LONDOrT M ATHKMATICAL SOCIETY. LONDON: TRUBNER & CO., LUDGATE HILL. 1888. \AU riiihts reaenvil.] ftS'~^ denotes that operation which compounded with S produces 1, i.e., identity). If S and T are not permutable, T' is difierent from T ; we say, then, that T' proceeds from T by transformation, and call T and T' associates withia the main group. In fact, T' will correspond with T in all essential properties, e.g. (as we see at once), it has the same periodicity. Now let T be replaced by the operations T^, T^ . . . T,. . . . of any sub-group. Then the same thing happens (as we apply each time the said *S^ to every T) to the corre- sponding T', so that, in fact, T\T\=- T\, when T^T^ coincides with T^. t We say that the groups of T and T' are then themselyes associates within the main group. We must consider now in particular the case where two different sub-groups (the original and the transformed) coin- cide with one another. If this occurs in the case of a set of operations, which we may choose from the entire group for the transformation of our sub-group, and if our sub-group thus shows itself only associated with itself, then we call it a self- conjugate sub-group. Every group contains, if we like to press the definition so far, two self -conjugate sub-groups: viz., in the first place, the totality of all its operations, i.e., the group itself, and, in the second place, that simplest group which consists of the identical operation alone. If a group contains, apart from these improper cases, no self -con jugate sub-groups, it is called simple, otherwise it is called composite. In the case of composite groups we seek especially their decomposition. We efiect a decomposition of a group by giving * If r= STS-\(T'f=STS-\ STS-'=Sr'S-'^; generally {T')'=STS-^ If, then, T"= 1, (2°)" = 1 also and conversely, q.e.d. t For we have again T'T^ = STS - '. ST,S - ' = STiTtS - ' = ST.S - \ 8 THE REGULAR SOLIDS a self-conjugate sub-group, as extensiye * as possible, contained in it ; then, again, a new one, self-conjugate within the sub- group so obtained, &c., and so on till identity is reached. It need hardly be said that this decomposition process admits of much variation according to circumstances. Beyond these simplest definitions, which come under our notice in the case of individual groups, I must consider that relation between two groups which is described as isomorphism. Two groups are called isomorphous if their operations can be so exhibited that S-^ S^ always corresponds to S'i S\ provided that Si is made to correspond to S'i and S^. to ^^. The isomorphous relation can be a mutually unique one ; we then speak of holohedric isomorphism. In this case the two groups, from an abstract point of view, are in general identical, and it is only in the significance of the two sets of operations that a difference can exist. The sub-groups of the one group, therefore, give directly the sub-groups of the other group, &c. , &c. But the co-ordination may also be an ambiguous one, and then we describe the isomorphism as merihedric. Here again to every sub-group of the S group corresponds one of the S' group, and vice versa, but the two sub-groups need not possess the same degree. At the same time, associate sub-groups of the one give similar sub-groups of the other. Therefore, also, self-conjugate sub-groups of the one group are transformed into similar ones in the other. In particular to identity, if we attribute it to the 5 group, corresponds a self-conjugate sub- group within the S' group and conversely ."f" In what follows we shall have principally to do with ex- amples of merihedric isomorphism, in which to each S corre- sponds one (Sf, but to every S' two /S's are co-ordinated (so that the number of the S's is double as great as the number of the S"s). We shall then simply speak of hemihedric isomorphism. § 3. The Cyclic Rotation Groups. Turning now to the closer consideration of the groups which are formed by the rotations which bring one of the configura- * That is, one which is not contained in a sub-group more comprehensive and at the same time self-conjugate. + Cf. besides the publications already mentioned, in particular : " Capelli, sopra I'isomorfismo . . ." in Bd. 16 of "Giornale di Matematiche " (1878). AND THE THEORY OF GROUPS. 9 tions mentioned in § 1 into coincidence with itself, we must give precedence to the simplest rotation groups, those which are obtained hy the repetition of a single periodical rotation. Evi- dently, for such a group, two points on our sphere remain unaltered, which points we will call the two poles; and the group consists, if it contains on the whole n rotations, of the n rotations through an angle = — — 2(w-1)t ' w ' K ' ' ' ' ' n round the axis joining the two poles. We agree in the first place that any two rotations of this group are permutable with one another. Therefore every indi- vidual rotation, as well as every sub-group which can be com- posed with individual rotations, is only associated with itself. But whether such sub-groups exist depends on the character of the number n. If to is a prime number, the existence of a proper sub-group is a priori excluded (because its degree must be a factor of n) • \i n is composite, there is, corresponding to every factor of n, one and only one sub-group whose degree is equal to this factor.* We shall obtain a decomposition of our group if we first seek the sub-group which in this sense cor- responds to the highest factor contained in n, and then further treat the sub-group thus obtained in the same way. If we like to familiarise ourselves with the idea of isomor- phism here directly, we observe that our group is holohedrically isomorphous with the totality of the " cyclic " permutations of any n elements taken in a definite order : (Ofl, (Zj, Oj a„_i). In fact, we can establish a correspondence between the per- mutations alluded to and the rotations which we have been considering most simply by geometrical means. We have only to construct the n points : ^01 '^l ^2» • • • • ^n— 1 which are derived from an arbitrarily given point a^ by our rotations, and now remark how these points are permuted amongst themselves by the rotations. * I make these and similar statements in the text without proof, because they will either be self-evident to the reader, or must be apparent to him, without further proof, on a little reflection. 10 THE REGULAR SOLIDS It is superfluous to spend more time over such obvious matters. We had to introduce them because the cyclic groups are, so to say, the elements from which all others are con- structed. § 4. The Group of the Dihedral Eotations. Turning now to the configuration of the dihedron, I beg the reader — here and in the similar developments of the following paragraphs — to make the corresponding diagrams, or to think out for himself directly by aid of a model — which is easily constructed — the properties under consideration. For we are treating of concrete matters, which may easily be conceived with the assistance of the suggested aids, but which may occa- sionally offer difficulties if these are neglected. I should also have had throughout to lay down these developments much more in detail, had I not wished to take for granted the reader's co-operation in the manner explained. We have already named that great circle on our sphere which carries the n summits of the dihedron the equator, and have also already marked the two corresponding poles. Then it is clear from the first that the dihedron is transformed into itself by the cyclic group of n rotations for which these poles remain unchanged. But the group of the rotations belonging to the dihedron is not thereby exhausted. We will mark a new point on the equator, midway between some two conse- cutive dihedral points ; the points so obtained we call the mid- edge points of the dihedron. We then further denote that diameter which contains a summit or a mid-edge point of the dihedron as a secondary axis thereof. There are n secondary axes of the dihedron ; if « is odd, each of these contains one summit and one mid-edge point ; if m. is even, the secondary axes separate into two categories, according as they connect two summits or two mid-edge points. In every case the dihe- dron remains unaltered if it is turned right round on any one of these secondary axes; i.e., if it is rotated through an angle TT round the secondary axis. Thus, by the side of the cyclic group of n rotations already explained, there are arrayed n other rotations, each of the period 2. Besides the rotations here enumerated the dihedron group con- AND THE THEORY Of GROUPS. ii tains no others. In fact, we recognise in the following way (which will be again applied later on) that the number of the dihedral rotations must be equal to 2«. We consider, first, that every point of the dihedron can be transformed into every other poiat by means of a dihedral rotation, which admits of n possibilities, and then that, while we keep one summit fixed, the dihedron can only be brought into coincidence with itself in two ways, viz., by a revolution on the secondary axis, which passes through the summit in question, and by the identical opera- tion. Now the number of dihedral rotations must evidently be equal to the product of the two factors ; it will therefore be equal to 2n, q.e.d. I will not now weary the reader by enumerating all the sub-groups contained in the dihedral group. Let us rather consider forthwith our first cyclic group of n rotations, and prove that this, as a siib-grotip within the main groiip of the dihedron, is self-conjugate. In fact, let us go back to the definition of § 2. We denote by T, T', rotations round the principal axis of the dihedron, and by S any other dihe- dral rotation. Then our assertion requires us to show that STS~ ^ = T'. But if S itself denotes a rotation round the principal axis, this relation is self-evident ; and if *S' is a revolution round a secondary axis, then the efiect of this revolution, so far as the principal axis is concerned, will be reversed by the operator S~^ following, from which our relation again results. We can refer the proof here given to a general principle, which we introduce here the more readily because in the sequel it will be repeatedly applied. Let as agree first that we will describe in our configurations all such geometrical figures as proceed from one another by an operation of the correspond- ing group as associates. We now construct all figures which are associate with a given one. Let T^ be those operations of our group which have the property of leaving unaltered every one of the figures so constructed. Then the T^ evidently form a self-conjugate sub-group within the main group. For every operation STiS~ ^ belongs itself to the T^, because *S^ only efiects a permutation of the fundamental figures, which will be reversed by S~^. The application of this principle to our case is clear. We have only to consider the two poles of the dihedron as 12 THE REGULAR SOLIDS fundamental associate figures. It is here incidental (as far as the general principle is concerned) that those rotations, which leave one of these poles unaltered, do not in general differ from those which transform both poles together into themselves. By similar reflexions we determine those among the dihe- dral rotations which are associated with one another. I say with regard to this, that now, of the rotations round the principal axis, those two which rotate through — and ^ are associates, while the revolutions round the secondary axes, for n uneven, are all associates, but, for n even, separate into two categories of associates. The first statement corresponds to the circumstance that the two poles of the dihedron are respec- tively equally affected by the two rotations round the principal axis which we are comparing,* the latter statement to the earlier theorem that the secondary axes of the dihedron are either all associates, or, for n even, divide into two sorts of associated lines. And further, in both cases we apply a general prin- ciple, which we can express by saying : Those two operations are always associates which transform respectively two associated figv/res analogously into themselves. I do not spend more time over the proof of this principle. If, finally, a decomposition of the dihedral group is required, such an one is already implicitly contained in what has gone before. As a sub-group at once the most comprehensive and self-conjugate, we choose the group of n rotations roxmd the principal axis. This we treat further ia accordance with the theorems of the preceding paragraph. We define another group of permutations of letters which is holohedrically isomorphous with the dihedral group. For this purpose we will now denote the n summits of the dihedron in their natural order by Then we have first, as in the preceding paragraph, corre- sponding to the n rotations round the principal axis, those cyclic permutations of the a/s which replace respectively a^ by (^p+k (*^^ indices being taken for the modulus n). We find, 2it IT * Inasmuch as a rotation through roxmd one pole coincides with a rota- n IICTT tion through H round the other. AND THE THEORY OF GROUPS. 13 further, that by a revolution round the axis which passes through the point a^, a, will be replaced by a„ _ ^. From both operations together springs the metacyclic group,* which will be represented by the following transformation of indices : v' = + » + A (mod. n), and this, therefore is holohedrically isomorphous with our dihedral group, or — ^what is the same thing — is identical with it in an abstract sense. § 5. The Quadratic Group. The explanations of the foregoing paragraph, as also the de- finition of the dihedron in § 1, assume that n is > 2. If w = 2, the figure of the dihedron loses its definite character, inasmuch as the suiomits of the dihedron can then be connected by an infinite number of great circles. In accordance with this we obtain, in the first place, as the corresponding group of rotations, a so-called continuous i group. Interesting and supremely important as the theory of the continuous groups is in many respects, it will be of little moment in the following pages. We will therefore, in the case of n = 2, make the figure of the dihedron definite by selecting from among the infinite number of great circles passing through the two summits a determinate one as equator. The principal axis of the figure then forms with the two secondary an orthogonal triad, and we obtain, in exact accordance with the rules of the preceding paragraph, a corresponding group of 2n = 4> rotations. If we make the usual determination of co-ordinates on the basis of this axial triad, the point x, y, z will be trans- formed by these rotations into the other points : * We thus denote generally with Kronecker every group of permutations of Oj, Oi, . . . It, which is given by »' ^ en + it (mod. n). + Of. the extensive investigations of Lie in the Norwegian Archiv (from 1873 onwards) and in Bd. xvi. of Math. Ann. Latterly M. Poincari, in his investiga- tions (which we shall often quote) of single-valued fimctions with linear trans- formations into themselves, uses the word " continuous group " in another sense. He describes as such every group of infinitely many but discrete operations, among which infinitely small transformations occur. This modification of nomenclature appears, however, to me to be not to the purpose. 14 THE REGULAR SOLIDS «. -y, -2/ -X, y, -z; -X, -y, z. Clearly our new group contains, apart from identity, only operations of period 2, and it is incidental that we have con- nected one of these operations with the principal axis of the figure, and the other two with the secondary axes. So I will give the group a special name which no longer recalls the dihedral configuration, and call it the quadratic group. The quadratic group has the special property, as is at once proved, that all its operations are permutable.* Thus every operation appears as only associated with itself, f We shall efiect the decomposition of the quadratic group by first descending to an arbitrary sub-group of 2 rotations, for which one of the three axes remains fixed, and then passing from this to identity. § 6. The Group or the Tetrahedral Rotations. We remarked above that, for all rotations which bring a regular tetrahedron into coincidence with itself, the counter- tetrahedron will also be transformed into itself. These tetra- hedra by their eight summits together determine a cube. If we now mark those 6 points on the sphere which corre- spond to the middle points of the sides, we obtain the 6 summits of a regular octahedron. We thus recognise already the close relation in which the group of the tetrahedral rota- tions stands to the octahedral group which we are now going to study. We will complete our figure by adding thereto the rectangular triad of the diagonals of the octahedron, and also the 4 cube-diagonals (passing through the centre of the sphere). Applying now the principles developed in § 4, we find, first, that the tetrahedral group contains 12 rotations. In fact, there are 4 associate tetrahedral points, and each of * It is easily shown that two rotations are only permntable if either (as in the case of the quadratic group) their axes cross at right angles and each has the period 2, or (as in the case of the cyclic group) their axes coincide. + This is not contradicted by the fact that, in the more comprehensive group to be now studied, the 3 rotations of the period 2, which the quadratic group contains, appear as equivalent. AND THE THEORY OF GROUPS. 15 these summits remains unchanged by 3 rotations — by the identical rotation, and by two rotations of the period 3 whose axes are respectively the cube-diagonals which passes through the tetrahedral poiat. We have ascertained at once, by what has been said, that 8 of our 12 rotations possess the period 3. Of these (again in virtue of the principles enunciated in § 4) four are associates, namely, all those sets of four which appear to rotate in the same sense, through an angle of -jr- (or "o- ), round the summit of the tetrahedron which they leave fixed. To these 8 rotations and identity are then added 3 more associated rotations of period 2. These are revolutions round the 3 mutually rectangular diagonals of the octahe- dron, which latter now appear as associates among themselves, because they are interchanged by each rotation of period 3. Together with identity, the 3 rotations in question evidently form a quadratic group. We conclude at once that the quadratic group so oltained is self-conjugate within the tetrahedral group. For the 3 mutually associated diagonals of the octahedron all remain unaltered for the rotations of the quadratic group, and only for them. We can therefore decompose the group of the tetra- hedron by first descending to the quadratic group, and then, treating this further in the sense of the preceding paragraph, I omit the proof that any other decomposition of the tetra- hedral group is not possible, and that generally, except the quadratic group, there exist within the tetrahedral group no sub-groups other than the simple cyclic groups which arise from the repetition of a single rotation.* Let us consider, further, the nature and manner of the permutations which the 4 diagonals of the cube (which we will shortly denote by 1, 2, 3, 4) undergo in virtue of the tetrahedral rotations. First, we have the self-evident asser- tion that by no one of the tetrahedral rotations (apart from identity) are all the 4 diagonals of the cube left unaltered. * Theoretically Bpeaking, we generate all the sub-groups of a given group by first constructing all the cyclic group mentioned in the text, and then combining these with one another in sets of two, three, &c., in order. In each individual case such a process can of course be considerably shortened by appropriate con- siderations. i6 THE REGULAR SOLIDS There are, therefore, no 2 tetrahedral rotations which gene- rate the same permutation of the four diagonals of the cube. Therefore the group of the tetrahedral rotations is holohedrically isomorphous with the group of the corresponding permutations of the diagonals of the cube* We see, in particular, that to the rotations of the self-con- jugate quadratic group correspond the following arrangements of the 4 diagonals : 1, 2, 3, 4 2, 1, 4, 3 3, 4, 1, 2 4, 3, 2, 1 To these are added, if we proceed to the remainder of the tetrahedral rotations, 8 more which arise from cyclic permuta- tions of 3 out of the 4 diagonals. We have thus, as we see, obtained just those 12 permutations of the 4 diagonals which we are accustomed to call the even permutations. § 7. The Gkoup of the Octahedral Rotations. In the case of the group of the octahedral rotations, we have, as has been already pointed out, essentially the same configuration for a foundation as in the case of the tetra- hedron. We will only further mark (on our sphere) the 12 points which correspond to the mid-edge points of the octa- hedron, and construct the 6 diameters which contain a pair of these points. These 6 diameters we call the cross-lines of the figure. Of course the octahedral group contains the 12 rotations of the tetrahedral group, and indeed, as we can premise, as a self-conjugate sub-group. For the 8 summits of the cube admit of being distributed between the tetrahedron and coun- ter-tetrahedron in only one way, and these latter remain both unaltered by the twelve rotations in question. Moreover, * Let us compare the behaviour of the 3 octahedral diagonals. These, since they remain unaltered for the operations of the quadratic group, are permuted by the 12 tetrahedral rotations only in 3 ways, viz., cyclically. With the group formed of these permatations, the tetrahedral group is then merihedricaUy isomoi- phous. AND THE THEORY OF GROUPS. 17 12 more rotations then arise which interchange the tetra- hedron and counter-tetrahedron, so that the octahedral group contaias on the whole 24 rotations. These are : first, 6 rotations — mutually associate — through an angle ir round the 6 cross-liues of the figure, than 6 rotations through — - (there- fore of period 4) round the 3 diagonals of the octahedron. The latter prove themselves also mutually associate. For the 4 rotations, for which the individual diagonals of the octa- hedron remain unmoved, now participate as a self -conjugate sub-group, tu a dihedral group of 8 rotations. Similarly the two rotations of period 3 round the same diagonal of the cube, and therefore in general all rotations of the period 3, are associate. For every diagonal of the cube is principal axis of a dihedral group of 6 rotations. The rotations of period 2, on the other hand, separate into two sharply defined categories, according as a diagonal of the octahedron or a cross-line remains fixed by them. The decomposition of the octahedral group is formed of course by descending first to the tetrahedral group, and then to the quadric group, &c., &c. No other kind of decomposition exists, as we have now ex- hausted in advance all the sub-groups contained in the octa^ hedral group. Finally, we agree that the diagonals 1, 2, 3, 4, of the cube, in virtue of the 24 rotations of the octahedral group, are per- mutated in 24 ways. The octahedral group is, therefore, holo- hedrically isomorphous with the totality of the permutations of 4 elements. § 8. The Group of the Ikosahedral Rotations. The group of the ikosahedron, to which we now turn, is for us the most interesting of them all, because, as we shall show, it is primitive, in contradistinction to the groups of the dihedron, tetrahedron, and octahedron. It shares this property with those cyclic rotation groups whose degree is a prime number. For the sake of iuvestigating the group of the ikosahedron, let us imagine that (in addition to .the 12 ikosahedral points), as we have said already, the 20 summits of the corresponding pentagon-dodekahedron (which correspond to the middle points i8 THE REGULAR SOLIDS of the sides) are constructed on the surface of our sphere, and, further, the 30 points which correspond on the sphere to the mid-edge points of the ikosahedron. The 12 ikosahedral points distribute themselves on pairs in 6 diameters, which we will describe shortly as diagonals of the ikosahedron. Simi- larly, corresponding to the 20 summits of the pentagon-dode- kahedron, we speak of 10 diagonals of the pentagon-dodeka- hedron, and finally of 15 cross-lines containing by pairs the mid-edge points. We convince ourselves first that the total number of ikosa- hedral rotations is 60. In fact, each of the 12 (evidently mutually associated) ikosahedral points remains unaltered on the whole by 5 rotations. We have thus at once (of course leaving the identical substitution out of the question), corre- sponding to each of the 6 diagonals of the ikosahedron 4 rota- tions of the period 5, in general, therefore, 24 rotations of this kind. In the sam^e sense the 1 diagonals of the pentagon- dodekahedron give 10 . 2 = 20 rotations of period 3, and the 15 cross-lines 15 rotations of period 2, whereby if we add identity the totality of the 60 rotations is exhausted : 24 + 20 + 15 + 1 = 60. Of the rotations here enumerated the 15 of period 2, and similarly the 20 of period 3, prove themselves respectively associate ; for they are the 15 cross-lines and the 10 diagonals of the pentagon-dodekahedron, and, if we rotate round one of 2v 4ir these diagonals through -q- or — , it comes to the same, so far as the main group is concerned, as if the two end points were again associated. On the grounds of similar considerations the rotations of period 5 are separated into two categories of 12 associates. The first category contains all rotations which turn through an angle of ± ^ round one of the diagonals of the ikosahedron, the other those whose angle of rotation amounts to ± -^• o With these data we have at once determined the cyclic sub- groups which are contained in the ikosahedral group. There are, as we see, 15 such groups having n = 2, 10 groups having AND THE THEORY OF GROUPS. 19 71 = 3, 6 groups having w = 5 ; cyclic groups with the same n are always associates. These data are suflBcient to prove the primitivity of the ikosahedral group. Namely, if a self-conjugate sub-group existed, this would have to contain either all or none of the cyclic groups having w = 2 (because these are associates), so too of the cyclic groups w = 3 or 71 = 5, either all or none. But the groups n=2, 3, 5, bring with them respectively 15, 20, 24, operations different from identity. If therefore we denote by 7}, t) , rf', three numbers which can represent or 1, the number of operations contained in the assumed self-con- jugate sub-groups amounts to : 1 + 15. )j + 20.V + 24.?i". But now this number, as we remarked before, must be a factor of the degree of the main group, and therefore of 60 ; this necessarily gives either : „^,' = V' = 0, whereby our sub-group coincides with identity, or : »i = l' = 'j" = l, which means that the sub-group is not distinct from the main group. The ikosahedral group is therefore primitive, q.e.d. Next to the cyclic sub-groups we find in the case of the ikosahedron, as a glance at the model teaches us, for further sub-groups, first, 6 associate dihedral groups having n — h and 10 associate dihedral groups having ?i = 3. The former have the diagonals of the ikosahedron, the latter those of the pen- tagon -dodekahedron as principal axes ; the corresponding secondary axes are contributed by the 15 cross-lines. We might suppose that in a similar manner, corresponding to the 15 cross-lines, 15 dihedral groups would present themselves having 71 =2, i.e., quadratic groups. Here, however, arises the fact that in the case of the quadratic group the principal axis is equivalent with the two secondary axes. In corre- spondence with this we obtain only 5 mutually associated quad- ratic groups. These correspond one by one to the 5 rectangular triads into which we can divide the 15 cross-lines. In these quadratic groups we have encountered that property 20 THE REGULAR SOLIDS of the ikosahedron which in the following pages will interest us most of all. Inasmuch as only 5 rectangular triads, as we have remarked, can be formed out of the 15 cross-lines, each of these triads must remain unaltered, not only by the rotations of the corresponding quadratic group, but on the whole for 12 ikosahedral rotations. It can he shown that these rotations f 01-711 a tetraliedral group. In fact, the 8 summits of the cube which corresponds to the rectangular triad to be considered are all included in the 20 summits of the pentagon-dodekahedron.* There are, therefore, contained eo ipso am^ong the ikosahedral groups those 8 rotations of period 3, which, together with the rotations of the fundamental quadratic group, form a tetrahedral group. We will also expressly agree that the 5 tetrahedral groups so formed are associates. Leaving again the proof that, besides those enumerated, no other sub-groups of the ikosahedral group exist, let us only further observe the isomorphism which arises in the case of the ikosahedral group from the existence of the aforesaid 5 rectan- gular triads. It can be shown that for every rotation of period 5 these triads are cyclically interchanged in a definite order. For each rotation of period 3, on the other hand, 2 of the triads remain unaltered, and only the other 3 are interchanged in cycle. Finally, it appears that for every rotation of period 2 one of the triads remains unaltered, while the other 4 are interchanged in pairs. In this manner it is shown that the group of the 60 ikosahedral rotations is holohedrically isomor- phous with the group of 60 even permutations of 5 things. We could of course here, as in former cases, have exhibited the essential isomorphism of our groups with certain groups of permutations of symbols, and then have transferred to the former the results which are found in the text-books with regard to the latter groups. Now we have investigated our groups directly, i.e., by means of the figures themselves, it will be a useful exercise to compare the results obtained by us vsdth the known properties of isomorphous groups. * One sees occasionally (in old collections) models of 5 cubes, which intersect one another in such a way that their 5 . 8 = 40 summits coincide in pairs, and represent the 20 summits of a pentagon-dodekahedron. AND THE THEORY OF GROUPS. 21 § 9. On the Planes of Symmetry in oor Configurations. For the further progress of our developments it is useful to construct the appropriate planes of symmetry of our configu- rations, i.e., those planes with respect to which the con- figuration is its own reflexion, and then consider the partition of the sphere which is effected by these planes. In the case of the dihedron, we can construct, besides the plane of the equator, n other planes of symmetry, viz., those planes which contain, besides the principal axis, one of the secondary axes. By means "of these (n + 1) planes, the sphere will be cut up into 4m congruent isosceles triangles, which have Iff IT ^ , 2 angles =^ and one angle = -. Of such triangles 4 meet in each dihedral point and in each mid-edge point, and 2n in each of the two poles, at equal angles. In the case of the regular tetrahedron, there exist 6 planes of symmetry, viz., those planes which, passing through an edge of the tetrahedron, are at right angles to the opposite edge. Consider for a m.oment a tetrahedron proper, limited by 4 planes, situated in space. Clearly each of the 4 equi- lateral triangles in these planes will be cut up by the planes of symmetry, through the agency of its 3 perpendiculars, into 6 alternately congruent and symmetric triangles. If we now transfer this partition by central projection on to the sphere, we have on the latter 24 alternately congruent and symmetric triangles, of which each exhibits the angles, x, ^. ^i and which at the summits of the original tetrahedron, as also at the summits of the counter-tetrahedron, meet in sets of 6, at the summits of the corresponding octahedron in sets of 4, with angles respectively equal. In the case of the regular octa- hedron, in addition to the planes of symmetry of the tetra- hedron, which as such are retained, 3 more arise : those planes which contain 2 of the 3 diagonals of the octahedron. By the 9 planes thus obtained the surface of the octahedron (which we will suppose for a moment to be a solid proper, constructed independently in space), consisting of 8 equilateral triangles, will be partitioned in quite a similar manner as the surface of the tetrahedron has just been. Passing by central THE REGULAR SOLIDS projection to the sphere, we obtain thereon 48 alternately con- gruent and symmetric triangles with the angles -^i -, ~, which meet at the summits of the octahedron in sets of 8, and at the ends of the cross-lines (the mid-edge points of the octahedron) in sets of 4. This is that partition of the sphere which is well known in crystallography in the case of the so-called Achtundvierzigflachner. Finally, in the case of the ikosahe- dron, we have, as planes of symmetry, those 15 planes which contain two of the six diagonals of the ikosahedron. These partition the 20 equilateral triangles which are contained in the bounding surfaces of the ikosahedron considered as a solid, exactly in the manner now several times considered. We obtain, therefore, 120 alternately congruent and symmetric AT AT T triangles on the sphere, whose angles are ^i -^t -^ and which meet in the summits of the pentagon-dodekahedron in sets of 6, in the summits of the ikosahedron in sets of 10, and in the ends of the cross-lines in sets of 4. Let us consider the similarity of the results thus obtained in the four cases. In each we have to do with a partition of the sphere iuto alter- nately congruent and symmetric triangles * which meet in sets of 2v in those points of the spherical surface which remaia immoved by a cyclic sub-group of v rotations. Of the num.- bers V there are in every case three, corresponding to the summits of the several triangles. They appear, in order of magnitude, collected in the following table, which may be kept in view during the later developments : «■! Co •■s Dihedron 2 2 n Tetrahedron . 2 3 3 Octahedron . 2 3 4 Ikosahedron . 2 3 5 * When we speak above, in the case of the dihedron, shortly of congruent triangles, no absurdity is involved, for we can, even in this case, describe the triangles as alternately congruent and symmetric, inasmuch as it is with isosceles triangles that we have to do. AND THE THEORY OF GROUPS. 23 We observe at once that the number of the triangles is in every case double as great as the degree of the corresponding group of rotations (which we will in future denote by N) ; they amount in the four cases respectively to 4w, 24, 48, 120. We complete these developments further by constructing, in the case of the cyclic groups also, certain planes which we call their symmetry-planes. These are to be simply such n planes, passing through the corresponding pole, as proceed from one another by means of the rotations of the group. These planes decompose the sphere into 2n congruent (or, if we prefer, alternately congruent and symmetric) limes, of angular separation -> of which each extends from one pole to the other. § 10. General Groups of Points — Fundamental Domains. We now apply the spherical partitions which we have obtained to the closer study of our groups of operations. We consider, first, the groups of points which arise if we submit an arbitrary point to the iV rotations of our group, and which we will call the aggregate of points or group of points belonging to our group of operations. Here we will suppose, for the sake of a clearer representation and more convenient description, the bounded regions on the sphere to be alternately shaded and not shaded. It is manifest a priori that for the rotations of a single group each shaded region will be transformed once, and only once, into every other shaded region, and similarly each non-shaded region once, and only once, into each non-shaded region. In fact, the number N of the rotations, as already remarked, coincides in every case with the half of the total number of regions. If, now, any point on the sphere is given (which may belong either to a shaded or non-shaded region), we can, thanks to our space-partition, without further trouble give the (iV— 1) new positions which it assumes in virtue of the (iV"— 1) rotations — distinct from identity — of our group ; we have simply to mark those (^— 1) points which are situated within the (iV"— 1) remaining shaded or non-shaded regions, in just the same way as the initial point in the original region. In general the iV points of the group of points so arising are all 24 THE REGULAR SOLIDS distinct ; they only partially coincide when the initial point retires to a summit of the -surrounding region. If, on the whole, V shaded (and of course the same number of non- shaded) regions meet at this summit, then the point will remain unaltered by v rotations of the group, and only assume N on the whole — different positions. The special sub-groups of points so arising are none other than those which we have otherwise considered in the foregoing paragraphs in our in- vestigations of the individual groups.* With the groups of points here constructed is connected a conception which will later on be of use to us. We describe as the fundamental domain of a group of point-transformations in general such a portion of space as contains one, and only one, point of every corresponding group of points.'\ The boundary points of such a domain are connected naturally in pairs by means of the transformations of the group, and only half of them can be attributed to it. I say now that, for our groups, we may consider as a fundamental domain in any case the com- bination of a shaded and a non-shaded region. In fact, if we allow a point to traverse a region thus defined without crossing its own track, the corresponding group of points cover uniquely the whole surface of the sphere. § 11. The Extended Groups. Applying the suggestions of § 1, we now extend the groups hitherto considered, by connecting with their rotations the reflexions of the respective configurations on the planes of symmetry. Here also the partition of the sphere given in § 10 will be of service to us. In fact, we recognise at once that each several region there distinguished, shaded or non-shaded, is a funda- mental domain of the extended group, and that therefore the extended group contains just 2N operations. As regards the * For the general groups of points mentioned in the text, consult the work of Sess already alluded to, where they are used for the purposes of the theory of polyhedra. t Cf. for different uses of this notion (so important for all applications of the theory of groups to geometry) my '^New Contributions to Riemann's Theory of Functions," in the xxi. Bd. Math. Annalen (1882). AND THE THEORY OF GROUPS. 25 proof of this statement, let us note, first, that a combination of the rotations hitherto considered with the reflexion on a single plane of symmetry suffices for turning each of our shaded regions into each of the non-shaded regions. On the other hand, let us reflect that a deformation of the sphere which is known to be a rotation, or to spring from a combination of a rotation with a reflexion, is completely determined as soon as we know that it transforms one of our regions into a definite one elsewhere. The fundamental domains so obtained have, in contradis- tinction to those considered in the foregoing paragraphs, the peculiarity of being in no wise arbitrary. In . fact, their boundary points are a priori determined by the fact that each remains unaltered by a determinate operation of the extended group, viz., by reflexion on a plane of symmetry. We can generate the extended group by connecting the initial group of rotations with the reflexion on that particular plane of symmetry in which the boundary point under consideration is contained. Therefore the special groups of only iV" points, which spring from the boundary points of the fundamental domains by the application of the extended group, are at the same time general groups of points in the sense of the fore- going paragraphs. Moreover, they are the only ones amongst these groups of points which at the same time remain unaltered by the operations of the extended group. Of course the special point-groups of — points just mentioned, correspond- ing to the summits of the fundamental domains, are also included among them. We might here have investigated our new groups, the ex- tended groups, in the same sense by the theory of groups, as we have done for the original groups in the preceding para- graphs. I should like to recommend such a discussion to the reader as an appropriate exercise, and limit myself here in this direction to the following statement : — ^The original group is in every case manifestly self-conjugate within the extended group. But, besides this, the extended octahedral and ikosa- hedral groups, as well as the extended dihedral group for n even contain a self-conjugate sub-group of only two operations. This springs from a double application of that transformation 26 THE REGULAR SOLIDS ■which replaces every point on the sphere by that diametrically opposite to it.* § 12. Generation of the Ikosahedeal Group. Hitherto, in our consideration of groups, we have supposed the individual groups ready to hand, and sought to obtain a uniform view of their different operations, and of the position of these latter with regard to one another. In the following pages, however, we shall find a more one-sided process of practical value. Our busiuess will be to introduce the groups by appropriate generative operations, i.e., to present operations from which, by repetition and combination, the group in ques- tion arises. We treat, first, in this sense the group of the ikosahedral rotations, here again taking advantage of the partitioning of § 9 and the fundamental domains of § 10 respectively. The principle, which here serves as our basis, has been already implicitly applied in the preceding paragraphs. Since each fundamental domain of a group will only be obtained from any other by one operation of the group, we can name the different fundamental domains after the operations, in virtue of which they proceed from an arbitrary one amongst them, which we will denote by 1 , as being the initial domain. Effect- ing this nomenclature, we obtain directly from it an enumera- tion of all the operations of the group, t We will suppose, for the sake of a more convenient mode of expression, that the ikosahedron is so placed that one of its diagonals runs vertically. For a first fundamental domain we then choose one of the 5 isosceles triangles, which, endowed with angles -^. 3. s' are grouped on the sphere round the * As especially remarkable, I will tidd that the extended octahedral group, consisting of 48 operations, contains 3 different self-conjugate sub-groups of 24 operations. These are first, as is manifest, the original octahedral group and tlie extended tetrahedral group, and then that group which consists in a combina- tion of the original tetrahedral group with the operation just mentioned in the text. Only the latter group, not the "extended" tetrahedral group, is a sub- group of the " extended " ikosahedral group. f Consult here the already mentioned " Grappentbeoretischen Studien " of Herr Dyck, in Bd. xx. of Math. Ann. The principle mentioned in the text is there applied to the general purposes of the theory of groups. AND THE THEORY OF GROUPS. 27 uppermost summit of the ikosahedron : such a triangle is a fundamental domaia of the ikosahedral group, because it is composed of two neighbouring triangles of the partitioning given in § 9. The five isosceles triangles in question form, we will say, a first pentagon of the pentagon-dodekahedron belong- ing to the ikosahedron. Those sides of a triangle which are at the same time sides of a pentagon we will describe as the (/round-lines of the figure in question. We now denote by S the rotation in a determinate direc- 2t tion through an angle, -^> round the vertical diagonal of the ikosahedron. Thus the 5 fundamental domains before men- tioned will proceed in their natural order from the first of them by the rotations: 1, S, S\ S^, S\ we will therefore denote the domains by the symbols S'^, Ac = 0, 1, 2, 3, 4. We now take a second ikosahedral rotation, T, of period 2. This shall be the revolution round that cross-line of the ikosahedron, one of whose ends is the mid-edge point of the ground-line of 1. By means of this T, our 5 domains ^ are transformed into the domains S''T, which, taken together, again make up a pentagon of our pentagon-dodekahedron, and, in fact, that one which has in common with the first pentagon just considered the ground-line of the first funda- mental domain. Applying now again the operations S, S^, S^, S*, we obtain from the new pentagon the remaining 4 attached to the first pentagon. Therefore, the fundamental domains of those 5 pentagons which surround the first one are represented by: S'^TS", (^, . = 0, 1, 2, 3, 4). A third ikosahedral rotation, also of period 2, shall now be denoted by IT, of which, however, we shall see that it has no independent importance, but is compounded of the two S and T. The axis of U shall coincide with one of the cross-lines which run horizontally, and, indeed, to make everything deter- minate, we will choose that horizontal cross-line in particular which stands at right angles to T. Clearly the rotation U so determined transforms the 6 upper pentagons of the penta- 28 THE REGULAR SOLIDS gon-dodekahedron which we have hitherto considered into its 6 lower pentagons, which were still wanting. Therefore we find at once that the thirty fundamental domains of the ikosa- hedral group which were still wanting are given by the following : S'U, S^TS'U, (fi., » = 0, 1, 2, 3, 4). From the fundamental domains we now turn hack to the rotations. We then have the proposition, the deduction of which was the object of our present considerations, viz., that tlie 60 rotations of the ihosahedral group are given hy the follow- ing scheme : 5'", ^TS", S^U, S^TS'U, (fi, » = 0, 1, 2, 3, 4). Here the rotations : form the dihedral group n = o belonging to the vertical diagonal of the ikosahedron, and the rotations : T, U, TU give, when taken together with identity, one of the 5 quadra- tic groups occurring in connection with the ikosahedron. If we draw a figure, as seems indispensable for the full understanding of the theorems here developed, or if we operate, as is more convenient, by means of a model of the ikosahedron on which the different fundamental domains are marked out and the corresponding symbols introduced, we can of course at once read off all the operations which make up any sub-group of the ikosahedral group. We have only to mark those fundamental domains which proceed from the domain 1 by the operations of the sub-group.* It remains for us to generate U, as we proposed, by a combination of >S^ and T. To this end we subject, say, the fundamental domain J^TSF to the operation T. Thus arises a fundamental domain *S^TO^^ which belongs to one of the pentagons of the lower half. But we have previously called * E.g., I find for the tetrahedral group which embraces the quadratic group just noted : 1, T, STS^ S'TS, ^-TS*, S'TS' V, TU, STS^U, S'TSU, S^TS*U, S^TS'U. AND THE THEORY OF GROUPS. 29 this same domain (as a glance at the figure shows) TS^U. Hence : ^TS^T=TS^U. In this equation let us consider U as the unknown. We solve this equation by first multiplying hy T on both sides of the left-hand expression and then by ^ and recalling that !P = 1, ^ = \. In this manner we have : U = S^TS^TS^T, and this is the relation we wanted. § 13. Generation of the other Groups of Eotations. As regards the generation of the other groups of rotations, this can follow without further trouble by the same means as we have now applied to the case of the ikosahedron. But for the first of these, the cyclic and dihedral groups, the matter is so simple that we need no special method, and for the tetrahedron and octahedron we propose in the sequel to use a method of generation which runs parallel with the decomposition of these groups before noted. I gather together here the results in question, which are easy to verify without special deduction. Now, as regards the cyclic groups, their operations will be manifestly given by the symbols : S^(^=0, 1,2, .. . (n-l)), where S denotes the rotation through the angle — . We obtain the group of the dihedron if we annex any revolution T round one of the auxiliary axes of the dihedron, and therefore add to the operations >S''' the others : ^^T, (^ = 0, 1, 2,. . . («-l)). In particular, the operations of the quadratic group are now represented (in agreement with the data just given) by the following scheme : \,S,T, ST. From the quadratic group we now ascend to the tetrahedral group by annexing any one of the corresponding rotations of 30 THE REGULAR SOLIDS, ETC. period 3, which we will call U. The 12 rotations of the tetrahedron will then be given by the following table : 1, S, T, ST U, SU, TU, STU, V\ SU\ TIP, STU^ Finally, we get the 24 rotations of the octahedral group on annexing to the 1 2 rotations here enumerated the others : V, SV, TV, STV, UV, SUV, TUV, STOV, VW, SIPV, TCPV, STIPV. Here V denotes any one octahedral rotation which is not con- tained in the tetrahedral group, e.g., a rotation of period 4 round one of the octahedral diag'onals. We here conclude these preliminary considerations. Their object was to instil into comparatively elementary geometrical figures the ideas of the theory of groups, in such a form that the group-theory reflexions and the geometrical mode of illus- tration might henceforward supplement one another. ( 31 ) CHAPTEE II. INTRODUCTION OF (x + iy). § 1. First Pkesentation and Survey of the Developments OF THIS Chapter. The essential step for our further progress in developing our train of thought is as follows : to consider the sphere which we submitted to the groups of rotations, &c., and on which we studied the corresponding groups of points and fundamental domains, as now the vehicle of the values of a complex variable z = x + i)/. This method of representation, originating with Riemann, and first thoroughly expounded by Herr C. Naiimann* in his " Vorlesungen iiber Riemann's Theorie der Abel'schen Integrale," is at the present day suffi- ciently well known, so that I can make use of it immediately ; besides, the formulas furnished in the following paragraphs are in themselves an efficient introduction to the theory. In virtue of the representation thus introduced, the indi- vidual system of points which we have hitherto considered appears defined by an algebraical equation f(z) = 0, where the degree of / is identical with the number of points, as long as none of these points retires to 2 = oo , which declares itself, in the well-known manner, by a fall of one unit in its degree. We inquire what properties these equations possess corre- sponding to the circumstance that the groups of points re- presented by them are transformed into themselves by certain rotations of the sphere, or by certain reflexions, &c. * £ieipzig, 1865. Cf. for the general application of Riemann'a method my treatise, " Uebec Riemann's Theorie Her algebraischen Functionen und ihrer Integrale " (Leipzig, 1882). Cf. again for the connection of this introduction of (x + iy) with the projective treatment of surfaces of the second order my work (still more often to be alluded to), " Ueber bintire Formen mit linearen Transfor- mationen in sich selbst," in Bd. 9 of Math. Ann. (1875), particularly at p. 189. 32 INTRODUCTION OF x + iy. Witii regard to this we have, first, the fundamental theorem, which I will presently establish and define more precisely, viz., that every rotation of the (x + iy) sphere on its centre will be represented by a linear substitution of z : (1) z'^""-^. yz + In fact, the z, which we can suppose extended with its com- plex value over the original sphere, and the z', which, in just the same way, we can suppose extended over the rotated sphere, are, in virtue of the interdependence of the two different spheres, related to one another uniquely without exception ; and, moreover, since the relation between the two spheres is one of conformity,* they are analytically re- lated to one another ; they are, therefore by known theorems, linearly dependent on one another, t So, too, we recognise that, to the reflexions and other inverse operations (which spring from the composition of a reflexion with arbitrary rota- tions), correspond formulae of the following kind : (2) 2'="it^ where z denotes the conjugate imaginary value (x — iy) of z. Our equations f{z) = have, therefore, the property of remain- ing unaltered by a group of linear substitutions (1), or, in some cases, by an extended group which contains, alongside of substitutions (1), a corresponding number of substitutions (2)4 * It is indeed one of congruency, since the corresponding points of either sphere can be brought into coincidence with one another by rotation. t Unfortunately we find the fundamental theorems of the function-theory, such as we are now considering, developed in the text-books in such a form that the conformable figure which is furnished by the functions is only incidentally taken into consideration ; it is, therefore, for our purpose necessary to make, in every case, a. certain modification and combination of the proofs explicitly given ; these, however, can present no difficulty to the reader, since we are always concerned with quite elementary relations. t- The same, of course, is true of equations Fiz) = 0, which, when combined, represent several groups of points such as are considered in the text. We can consider these equations F{z) = as a generalisation of the reciprocal equations of lower analysis, inasmuch as the latter also remain unaltered by a definite group of linear substitutions, viz., by the simple group i=z,i!= -. z INTRODUCTION OF x + iy. 33 I mnst now consider at once the analytical method which occurs spontaneously in the establishment of the equations /(«) = and in the study of their mutual relations, and which, by virtue of its more varied aspects, excels in many respects the former reflexions based on geometrical illustrations : that of the koTiiogeneotbs variables. If we replace z by Zj : 2 , the substitution (1), (and analogously every substitution (2)), splits up into two separate operations : (3) 2', = '«Z1+)SZ2, ^2 = 7h + *«2. where now the absolute value of the determinant (aS — ^y) of the substitution will be of especial importance. Instead of the equations f(z) = or /( -i, 1 j =0, we shall then have to consider the form f{z^, z^, on multiplying by a proper power of z^. This form has always the same degree (a first point in favour of the homogeneous notation) as the corresponding group of points, the occurrence of the point z = co being now indicated by a factor z,^ of /. We recognise at the same time that, with the transition to a form, /, a new distinction arises. For / need not remain absolutely unaltered for the substitu- tions (3) ; it can change to a factor pres, and our business will be to determine this factor. Moreover, we obtain, by putting in the foreground the consideration of the theory of forms, a bond of union with that important discipline of modem algebra which is described as the theory of invariants of binary forms. This will be of service to us in the more complicated cases, in order to deduce from one form / all the rest in a simple manner. I may mention at once the result in which the considerations here explained culminate (see the paragraph before the last of this chapter). It is this ; that for each group of linear substi- tutions (1) corresponding to our group of rotations, a corre- sponding rational function : (4) Z=R{z), will be found, which represents the different groups of points belonging to the group, if we equate them to a parametric constant. But at the same time we obtain, if we actually represent those groups of substitutions, a series of new prob- 34 INTRODUCTION OF x + iy. -lems, from which, later on, our further development will have to start.* § 2. On those Linear Transformations of (x + iy) which Correspond to Rotations Round the Centre. Let the equation of our sphere, relatively to a system of rectangular central co-ordinates, be (5) g2 + „2+^2=l. We then introduce the complex magnitude z = x + iy, by first exhibiting (x + iy) in the usual manner in the ^17-plane (the equatorial plane), and then, placing this plane by stereo- graphic projection ffom the pole ^=0, »; = o, ^=1, ina(l, 1) relation with the surface of the sphere. We thus obtain the formulae : (6) a;=j^. t/=j^, a; + i2/ = y^^-; or : (") ^-1+3.2 + 2,2.1-1+3.2^2',- l+a^ + y2 As we particularly want to determine these linear substitu- tions of z which correspond to the rotations of the sphere, the diametral points of the sphere are of interest to us (inasmuch as one pair remains unmoved in every rotation). In order to derive, with reference to these, a preliminary theorem, we substitute in (6), instead of ^, rj, ^, their negative values. Then we have for the diametral point : and therefore, by multiplication with the values (6) of (x + iy) and attending to (5) : * Consult throughout the work already mentioned, " Ueber binare Formen mit linearen Transformationen in sich selbst," in Bd. 9 of Math. Ann. (1875). It is there that for the first time that process of thought is displayed from its founda- tions which now reaches a detailed exposition in the developments of the first and second chapters of the text. I had communicated the principal results in June 1874 to the Erlanger physikalisch-medicinische Gesellschaft (c/. the Sitz- ungsberichte). INTRODUCTION OF x + iy. 35 (8) (x + iy)(x'-iy')^-\, or, if we put (x + iy) = re**, (9) y + i2/' = l. e< (*+")■ Diametrally opposite points have arguments whose absolute values are recipi-ocal, while their amplitudes differ hy tt. We -consider now, first, the case where the axis — 00 (which stands at right angles to the plane of the equator) is rotated through an angle a, and let this rotation, looking from the outside on to the point 00 (which we suppose placed on the upper side of the equatorial plane), take place in a sense opposite to that of the hands of a clock. A point, which ori- ginally had the argument z, will, after the rotation, have the argument /. We inquire how / is connected with z. Evi- dently in the same way as (^' + i>?') with (^ + iy), if we rotate the ^»7-plane (the equatorial plane) in the way given ; for the denominator (1 — ^ in the formulae (6) remains unaltered by the rotation. But now we have for the said rotation of the ^>;-plane, if, as usual, we let the positive ^-axis extend to the right, and the positive 17-axis away from us : §' = § . cos a - J) . sin a, »l' = § . sin a + >i . cos o, or, g' + ill' = (cos a +i sin a) (§ + in) ; whence follows in the well-known manner : '•■ (10) s' = e'».2. If we now wish to represent analogously a rotation through an angle a, for which the points ^, 17, ^, and — ^, — »7, — ^, on the sphere remain unmoved, and for which the first point plays the same part as the point 00 did before — so that, therefore, if we view f, >;, X., from without, the rotation takes place in a sense opposite to that of the hands of a clock — we have in (10), instead of 2 and J, such a linear function of z and / respectively as becomes infinite at ^, 17, ^, and vanishes at — ^, — >?, — X,- Such a liaear function is, however, determined, save as to a factor ; it runs in the most general form : 36 INTRODUCTION OF x + iy. S + iri G 1-^ But it is unnecessary to determine this factor more precisely by any kind of convention, because it must of itself drop out of the formula to be established. In fact, we obtain, on substi- tuting iu (10) for 2 our new expression, independently of C: ^ "^ 1 + r . ^'^ i + r~ , f + ^1 t + tn or, after an easy alteration : (11) vr ^'(^ + + (^+^i) 4- ^(^ + D + (g+»"'i) '^'(l-O-i^ + in)-' •z(l-?)-($ + z,)- This is there/ore the general formula for an arbitrary rotation, for which we sought. If we solve it for /, it will be con- venient to introduce the following abbreviations : (12) gsin^ = a, >isin^ = 6, ^sin" = c, cos^-=d, where evidently : (13) a2 + Z>2 + c2 + c£2=l. We then obtaiu the simple form : < ' _ (<^ + »c) z — (b- ia) * I I j (b + ia) z + {d- ic) We have, as we might suppose a priori, obtained by this method two formulm for every rotation of the sphere. The rotation remains unaltered, namely, if we increase the angle of rotation a by 27r. Now the consequence of this is, by formula (12), that all 4 magnitudes change their sign. This corresponds to the circumstance that the determinant of the substitution of (14) will be equal to a^ + ^l^ + c^ + c^^ therefore by (13) equal * See the note by Caj-ley in Bd. 15 of Math. Ann. (1879), "On the Corre- flpondence of Homographies and Kotations, " where this foriuula is for the first time explicitly established. ( INTRODUCTION OF x + iy. 37 to 1, which in respect to the sign of a, b, c, d, admits of just two possibilities. At the same time we have obtained a convenient rule for calculating the cosine of half the angle of a rotation which is given in the form and thereby estimating the periodicity of this substitution (so far as we are concerned with periodic substitutions). For manifestly we have, by comparison with (14) : (15) ^ cos ^ = . 2 J AD - BG § 3. Homogeneous Linear Substitdtions — Theie Com- position. We will now, as we proposed in § 1, split up formula (14) into two homogeneous linear substitutions by simply writing : (16) j"> = |f + !''=K-(*-*?^' ^ ' ( z 2 = (6 + ia)Zi + {d~ ic)z2. Here a, h, c, d, denote, according to formula (12), in the first place, arbitrary real magnitudes, which are subject to the condition : a2+J2 + c2 + ti2=l. Meanwhile we may remark that the same formula, with this condition maintained, provided we regard a, b, c, d, as susceptible of arbitrary complex values, represents at the same time the most general binary linear substitution of determi- nant 1 . Hereby the formulae of composition, which we shall immediately establish, acquire a more general significance, which, however, in the developments to which we must here limit ourselves, need not be further considered. To deduce the formulae of composition in question, let ir. {d + ic)z.^ - (6 — Ja)z2, (6 + m)zj + {d- ic)z^, be a first substitution, and similarly ^ ( z'\ = {d' + ic')z\-{h' -ia')z'^, ^ \ z\ = {h' + ia')z\ + (df + ic')z\, a second. We obtain the substitution ST, arising from the 38 INTRODUCTION OF x + iy. composition of these, hy eliminating J^, /j. from the two systems. We naturally put the result again in the form (16), and so write : ^^ \ z\ = (6" + m")zi + (d" - ic'y^. Then direct comparison gives the following simple result : (17) (a" = (ad' + a'd) - [be' - b'c), )b" = {bd' + b'd)-{ca'-c'a), ■) c" = led' + c'd) - (aV - a'b), Vd"= -aa' -bV -cc^-dd'. We have thus, as we may observe, the symbolic notation ST applied in the same sense as in the preceding chapter, if we effect first the substitution S, then the substitution T. We shall immediately apply the formulae (14), (16), (17), in the establishment of the groups of substitutions which now correspond to the groups of rotations of the preceding chapter. First, however, we must consider the significance which these formulas claim in a more general sense. That it was proper, in the treatment of rotations round a fixed point, to introduce the parameters a, J, c, A, of the preceding paragraph (or at least their quotients -5. ^> ^j, Euler had already found.* It appears, however, that the formulje of composition (17) re- mained still unknown for a long time, till they were discovered \)j Rodrigues'l (1840). Hamilton then made the same formulfe the foundation % of his calculus of quaternions, without at first recognising their significance for the composition of rotations, which was soon brought to light by Cayley. § But the relation of these formulae to the composition of binary linear substitu- tions remained still unobserved ; to Herr Laguerre is due the * " Novae Commentationes Petropolitanse," t. 20, p. 217. t " Journal de LiouvilU," 1. aerie, tome v. : "Des lois g^om^triques gui r^gis- eent le deplacement," &c. i In fact, if we consider the quaternions : q = ai + hj + ck +d, q'= a'i + Vj + c'k + d', the product thereof : qq'= 5"= a"i + b"j + c"k + tf is exactly given by the formulae (17) of the text. It is interesting to consult the first reports of Hamilton on hia calculus of quaternions, especially his letter to Graves in the " Philosophical Magazine," 184i, ii. p. 489. § "Philosophical Magazine," 1843, i. p. 141. INTRODUCTION OF x + iy. 39 credit of having first recognised this connection on the formal side.* It first acquired a real importance by Hiemann's inter- pretation of (x + iy) on the sphere, and especially by Cayley's t formula (14). § 4. Eetukn to the Gkoups of Substitutions — The Cyclic and Dihedral Groups. We now proceed to establish the homogeneous linear substi- tutions of determinant 1,J which correspond, in the sense of formulae (14), (16), to the groups of rotations previously inves- tigated. Of course, the substitutions which we iu this manner obtain are, on account of the double sign of the parameters a, h, c, d, dovMe as numerous as the rotations from which we start. The group of substitutions is, therefore, in the first place hemihedrically isomorphous with the group of rotations ; the question whether we cannot so limit or modify the group of substitutions that holohedric isomorphism ensues, will not be investigated till a later paragraph. As regards the general rules of which we shall make use in establishiag the groups of substitutions, we shall of course, in each case, provide for the system of co-ordinates a position as simple as possible ; and besides this, we shall recur to the propositions which we have established iu ^ 12 and 13 of the preceding chapter, with reference to the generation of the several groups of rotations. In the case of the cyclic groups and the dihedral groups, the afiair is so simple that we can write down the formula without more ado. It seems most convenient to let the two poles considered in connection with these groups coincide with the points 2 = and z= ca. Then we have, for the rotations of the cyclic groups : • "Journal de I'icole poly technique," cah. 42 (1867): " Sur le calcul des syst&mes lin^res.'' t Of. especially, too, M. Stephanos' article, " M^moire sur la representation des homographies binaires par des points de I'espace avec application i. I'^tude des rotations sphdriques," Math. Ann., Bd. xxii. (1883), and also his note "Sur la th^orie de quaternions " (ibid.). t Or, as I shall say in future for brevity, where there is no fear of misunder- standing: the "homogeneous substitutions." 40 INTRODUCTION OF x + iy. = 0, c = sm -, a = cos -, a = , ' 2 2 n and therefore for the 2re homogeneous substitutions of the cyclic group : ihr — ikr (1^) z\ = e% ^1, 2'^= e"^^. 22 . (/.■ = 0, 1, . . . (2« - 1)). If we now proceed to the dihedral group, we shall choose one of the secondary axes, so that it coincides with the ^-axis of our co-ordinate system in space (and therefore joins the points s = + 1 and 2 = — 1 on the sphere). We find for the corresponding revolution : (19) Z'l = + ^2, Z'g = + Wi, and therefore, by combination with (18), for the 47i homo- geneous substitutions of the dihedral group : ihtr — ikir = e . 2i, 2» = e . Zj; (20) ( . ., (A = 0, 1, . . . (2n-l)). = ie " .Zj, 2'2 = ie " . z^. Account has already been taken in these formulae of the double sign of (19), since we have allowed k to range, not merely from to (n — 1), but from to (2w — 1). We have in particular, as we will note expressly, for the quadratic group the following 8 homogeneous substitutions : (z\= i" .z^, z'2 = (-j)*.z2; (21) Vi=-(-i)*.S2. ^'2= i'.h; (i = 0, 1, 2, 3). § 5. The Groups of the Tetrahedron and Octahedron. In the case of the tetrahedron and octahedron we shall dis- tinguish two different positions of the system of co-ordinates. In the first case we allow, as appears most natural, the 3 co- ordinate axes ^, jj, ^, of our co-ordinate system in space to simply coincide with the diagonals of the octahedron. In the second case we rotate the co-ordinate system so obtained on its ^-axis through 45°, viz., so that (as proves advantageous later INTRODUCTION OF x + iy. 41 on) the ^^-plane coincides with a plane of symmetry of the tetrahedron. Let us begin with the consideration of the /ormer position. "We can then make immediate nse of formulae (21), just written down, for the representation of the quadratic group. Recall- ing now, with regard to the generation of the tetrahedral and octahedral groups, the data which we have prepared in § 13, we will first construct the homogeneous substitutions which correspond to the two rotations (V and U") of period 3 round one of the diagonals of the corresponding cube. Evidently 2 diametrically opposite summits of the cube have the co- ordinates : ?=,=c=±Jr and since : w 1 cos - = - = 3 2 2r . IT Js' 2ir -cos g.smg. 2 _sm 3, we obtain for the homogeneous substitutions for which these two summits remain unmoyed (neglecting the double sign, which occurs again here) : a = b = c = ± d = -=• Corresponding to this we have the two substitutions : , (±l+i)2l-(l-OV y _ ( 1 + i)h + ( ± 1 - ih zy, = 2 ^ 2 Combining these now in a proper manner with the substitu- tions (21), we obtain, for the right sides of the 24 homogeneous tetrahedral substitutions, the following pairs of linear ex- pressions : - ( - 0* • ^2, «'* ■ h ; * ( ± 1 + «>i - (1 - i)02 / _ -.k ( l+»>i + (±l-»)g , — 2 ' ^ ' ' 2 (l+lX + (+l-t>2 -s (±Jj^Zj-(lj-£)z2 ' 2 ' 2 ' k = 0,l,2,3) We pass to the octahedral group by adding a rotation V through 5 round one of the 3 co-ordinate axes, say the ^-axis. 42 INTRODUCTION OF x+hj. For one of the two corresponding homogeneous substitutions we have manifestly : (23) ,'^ = _i^.,^, ,'^=i^\,^. In correspondence with this, we obtain the right hand sides of the 24 homogeneous substitutions still wanting in the table (22), &y multiplying, in each case, the left-hand one of the 24 linear 1 +i expressions included in this table hy ~, — , the right-hand one by v2 1-t n/2-' It will be unnecessary to write down specially the new expressions here. With regard, now, to the second position of the co-ordinate system relatively to our configuration, it is sufficient, in order to have substitution formula with reference to it, to take account — in the formulae (22), (23), &c., just obtained — of the transformation of co-ordinates which leads from the first posi- tion to the second. For such a transformation of co-ordinates, the original — will be replaced by — 1^= ■ -> and of course simultaneously the original — ^ by • -J.* Let us observe, moreover, that —7= • —7^=1. We thus obtain on brief re- »/2^ \/2 flexion the rule : If we desire substitution formula which correspond to the new position of the system of co-ordinates, we must, in the expressions occurring on the left-hand side in (22), leave z. 1 —i unaltered and replace 2 by — t= ^2 j on the other hand, in v2 the expressions occurring there on the right-hand side, we 1 + 1- must replace z^ by -j= ■ h> and leave z unaltered. With such entirely elementary operations I agaiu omit to explicitly note the expressions which occur. * Namely, if we suppose the rotation through TOS^ound Of -axis to proceed iq a positive sense. INTRODUCTION OF x + iy. 43 § 6. The Ikosahedral Group. We have now to investigate the homogeneous substitutions of the ikosahedron. With this object we will endow the ikosahedron with such a position with regard to the system of co-ordinates that the rotation through -^> which we previously (§ 12 of the preceding chapter) denoted by S, takes place in a positive sense round the ^-a3s;is, while at the same time the cross-line, round which the revolution U {loco cito) takes place, coincides with the 17-axis. Then we have at once, corresponding to the operations S, U, the following substitutions : (24) \ \^2=±'\> \ U: I'rZ '^' which, taken together, generate the dihedral group belonging to the vertical diagonal of the ikosahedron.* By e here, as always for the future, the fifth root of unity : (25) . = « — is to be understood. Our convention respecting the position of the system of co-ordinates admits of a twofold possibility with regard to the revolution T, which we have now still to consider. The axis of T can move in the ^^-plane either through the first and third quadrants of the system of co-ordinates ^^, or through the second and fourth. We will settle that the latter is the case. If we understand by y the acute angle which the said axis makes with 0^, one of its ends will have the co-ordinates : g= -sin 7, 71 = 0, ? = cos 7, and the parameters of the corresponding rotation become by (12) (since we are concerned with a rotation through 180°) : a= +sm y, b = 0, c= ±cos y, d = 0; * This is here related to a somewhat different system of co-ordinates to that of formula (20). 44 INTRODUCTION OF x + iy. where, as always in these fonmilse, the upper and the lower signs go together respectively. The question now is how we calculate the angle j. For this pui-pose I will return to the parameters of S (24) : o' = i' = 0, c'= +sin -, d'= ±cos : 5 5 and to the formulae of composition (17). By means of this formula we find for the parameter d" of the operation ST : d"=-aa:-hV-cd + dd' . . v = +COS y ■ sm -. 5 Now the operation ^S"^ (as a glance at the figure of the ikosahedron shows) has the period 3 ; it must therefore be IT 1 identical with ± cos -= ±-. "We thus obtain, if we further consider that cos y must be positive : . T 1 cosy, sin 5 = 2' or, if we again introduce the root of unity e, and recall that we must have : COS y = -^—r=f and, therefore, again assuming the positive sign, ,2 - |3 em 7 = ■■JV We now introduce these values into the expressions a, h, c, d, just given, and also refer to the formulae (16). Then we have, finally, for the two homogeneous substitutions which correspond to the rotation T : (2^) lV5.2>±(.^-.^)^l±(s-.*)22. From (24), (26), we now construct at once the whole set of ikosahedral substitutions. We need only remember that we INTRODUCTION OF x + iy. 45 previously brought the ikosahedral rotations into the following table : Si--, S^U, &^TS', S'-'TS'U, (m, ^ = 0, 1, 2, 3, 4). Corresponding to this we obtain for the 120 homogeneous ikosahedral substitutions : S^TS' ■ j ^^- ''' = ± ''"(-(' - ^"') ^"* • h + i^' - ^^^ ■ -2), (27) { ■lV5.z'2=±f '{-(f - £4)63" . Zj + (i2 _ ,3),2'' . 22). I will further call attention to the simple rule by which here (as also in the previous cases), the periodicity of the individual rotation is determined by formula (15). We obtain, in virtue of this formula, for the angle a of a rotation S'^TS" : COS 2-+ ^2-^= and analogously for the angle of rotation of S^TS"!!: cos - = + ! '—^ '. 2 ^ 2jr \ We have, therefore, for S^TS^ the period 2, if m+>'/=0, for ^S^r^S^t/'if 3ju + 2)/ = (mod. 5). We have for S^TS" the period 3, if m + 1/ s ± 1, for S^TS'U, if 3m + 21/ s 1 ± (mod. 5). In the 20 other cases S'^TS" and ^TS^U are respectively of period 5. To this must be added, what is self-evident, that all the /S" U have the period 2 ; all the /S", with the sole exception of S° (identity), have the period 5. 46 INTRODUCTION OF x + iy. § 7. Non-Homogeneous Substitutions — Consideration of THE Extended Groups. From the homogeneous substitutions we naturally descend without calculation to the non-homogeneous substitutions. If I give here, notwithstanding, the formulae in question in a tabular collection, it is because, when the fixed value of the substitution-determinant hitherto maintained is abolished, they admit of a certain amount of condensation, and hence, in fact, become very readUy surveyed. We find for the non-homoge- neous substitutioTis : (i.) For the cyclic group : iikw ^2^)- 3' = e"^.2, (A = 0, 1 . . . («-l)); (ii.) For the dihedron : (29) e n z' = e . z, z = , (A; as before) ; z (iii.) For the tetrahedron and first assumption respecting the position of the system of co-ordinates : ,-. , , , .1 ,. z+1 ,. z— 1 ,z + i ,z—i (30a) \ z = ±z, ±-, ±1 . -, ±1 . -, ± ^ ± ;, "^ ■' z z-1 z+1 Z — l Z + l also, for the other assumption : (oOb) z = ±z, ± -1 ± -7= ; ) + -/T ■■ — ^^ — F-> ^ ■' z ^2. z-(l-i) '- (l+»)z+ V2 (1 -i)z+ J2 ^ J2~. z -{\+ i) . - J2.z-{l+i)' ~ ~(l-i)z+ ^2' (iv.) For the octahedron with similar distinctions in the two cases : /oi \ / •«• «* * z + 1 .1 z— 1 * Z + l .t z — i ^ ' z z- 1 z+I Z — l 2 + » and : (31b) z'=.-*.z.^,.-*.iLi!)l^. ^^_^;f_■i.-(l^i). '^ ^ z ^/2^■2-(l-^) {l+t)z+ J2 ^ {\-i)z+ J2 ,k V^'-.z - (1 + i) V ' ^/2■.^-a+»■)' . ' (1-0^+ V2J INTRODUCTION OF x + iy. 47 Tc has here in each case to assume in succession the values 0, 1, 2, 3. (v.) For the ikosdhedron : (32) z' = t-z,Z^,c. {t^-i^yz + {i -i*f (|2_,3y.2+(j _e4) -(i -i*y.Z + {t--l^)' V = e'; /*, »,= 0, 1, 2, 3, 4). From these formulae we now pass at once to those which correspond to extended groups (as we expressed it in Chapter I.), namely, if we deduct the single groups of formulae (30a), the ^t-plane is throughout a plane of symumetry for the configura- tion just considered. Now we can generate the extended group by combining the reflexion on this very plane of sym- metry with the rotations of the original group. This reflexion is, howeyer, given analytically by the formula : (33) ■ z'=~z, where z denotes the conjugate value of the imaginary quan- tity z. Hence we shall obtain formulce for the operations of the extended group if we place alongside of the formulce (28) to (32) ((30a) alone excepted^ the others in which z is replaced hy z. I conclude this paragraph with two short historical remarks. Of the groups of substitutions (28) to (32) only two cases come particularly into prominence in earlier literature (except the cylic groups, which, of course, occur everywhere), viz., the dihedral group for m = 3 and the octahedral group (31a). The first case appears in a form somewhat difierent to that of (29), but only because a difierent system of co-ordinates is established on the z-sphere, viz., that for which that great circle which we have hitherto described as the equator coin- cides with the meridian of real numbers, and the summits of the dihedron have the arguments 2 = 0, 1, cx5. We thus find the formulae : ,_ 1 , 1 Z 2- ] Z —Z, - , i.—Z, , Z-, > z \ -z z—\ ■ z which in projective geometry connect the 6 corresponding values of the double ratio and in the theory of elliptic func- 48 INTRODUCTION OF x + iy. tions (which is really the same thing) the 6 corresponding values of k^ (the square of Legendre's modulus). The group (31a) is found in several places in Abel's works.* The object is there to present the different values of h^, which result on transforming a given elliptic integral of the first kind by a linear substitution into Legendre's normal form : dx !-. Abel remarks that these different values are represented in terms of any one of them in the following manner : J. 1 /1+s^Y ^ I-n/^ V (i+JkV (i- JIY If we here extract the fourth root and replace ^J- by z all through, these are evidently exactly the expressions (31a). § 8. HoLOHEDRic Isomorphism in the Case of Homo- geneous Groups of Substitutions. For a discussion of the groups of substitutions now obtained from tie point of view of the theory of groups, it will be suflELcient to refer here to the analogous inquiries in our first chapter. In fact, our non-homogeneous groups of substitutions are holohedrically isomorphous with the groups of rotations there considered, the homogeneous ones at least hemihedrically, where let us expressly remark, that among the homogeneous substitutions the two : ^'~^' ' andV~ ~^^ always correspond to " identity." Moreover, we will concern ourselves with a question of an allied nature, certainly, if not purely one belonging to the theory of groups, a question which we have already pointed out (^ 4 supra), and the answering of which will be of prime importance to us in the sequel. We have found for a group of N rotations in every case 2iV homogeneous substitutions. We ask if it be not possible to extract from among these 2N substitutions iV^ of them forming a group so that holohedric * See, e.g., Bd. i. p. 259 (ne* edition by Sylow and Lie). INTRODUCTION OF x + iy. 49 isomorpliism with the group of rotations ensues, or if we cannot at least attaia that isomorphism, by impartiag any- other value to the determinant, which we have hitherto taken 4- 1 , of the individual substitutions ? We begin with the repetitions of a single rotation, i.e., with the cyclic groups, where, in order not to apparently limit the investigation by the introduction of a canonical system of co- crdiuates, we will start from a perfectly arbitrary system of co-ordinates. We therefore take, say, a rotation through — , for which an arbitrary point ^, »?, ^, on our sphere remains un- moved. To the corresponding linear substitution (16) : z\ ={d + ie)zj^ — (6 — ia)z.2, z'2 = (5 + ia)Zj^ + (d- ic)z2, we have hitherto attached the parameters : a= ±.t sm -, 0= + ri sin -, c = + l sm -, a = ± cos -. n n 11 n We will now write instead of them, taking the determinant of the siibstitution equal to p^ : (34:) «i = p2 sin -, b-, = on sm -, c,= p!' sin -,di = p cos -. ^ ■' ^K n n n Recurring then to the formulae of composition (17), we obtain for the parameters of the k^^ repetition of our substitution : Ot = p^.gsm — , 6i = p*. Jjsm — , c^ = p''. ; sm — , rfj, = |o* cos — n n n n We require now — in order that holohedric isomorphism with the corresponding group of rotations may take place — that the n^^ repetition of our substitution should be identity, and that, therefore : an = &n = c„ = 0, rf„=l. It is clearly necessary for this that : p"=-l. We shall, therefore, then, and only then, attain to holohedric isomorphism between the substitutions and the group of rotations when we introduce in (34) p as the n^ root 0/ (- 1). Hereby, however, the value p^ of the determinant of the substitutions is determined, or at least limited to a few possibilities only. If n is odd, we can take p= —\, and therefore the deter- minant = + 1. If TO is even, the value + 1 of the determinant 50 INTRODUCTION OF x + iy. of the substitution is iiiCTitable. In particular, if n = 2, we must choose the determinant = — 1, and the magnitude p= ±i- We now consider the dihedral group. We have for it, first of all, the rotations &^ (with S" = 1), which, according to what has just been said, we must make correspond to substitutions of determinant p^, where /o"= — 1. We have further the rotations S^T of period 2. To effect holohedric isomorphism we shall certainly provide the substitution which coiTesponds to T with the determinant ( — 1). Now we know that in the composition of two substitutions their determinants are multi- plied. Therefore we obtain for S^T a substitution of deter- minant — p^- But this must itself again be equal to —1, because S^T has the period 2. Thus we have for p the simul- taneous equations : p''=-l,/,2''= +1,(^ = 0,1. . . (n-\)). These are evidently only reconcilable when n is odd (whence p= — !)• Therefore it follows that, in the case of the dihedral group, the desired holohedric isomorphism can only exist for n odd, never for n even. We shall in the sequel lay special stress on the negative part of this proposition, for we at once deduce from it an analogous theorem for the groups of the tetrahedron, octa- hedron, and ikosahedron. In the case of the tetrahedron, octahedron, and ikosahedron, holohedric isomorphism between the grotip of rotations and the group of homogeneous suhsti- tutions is impossible. They all contain, namely, as sub-group at least one dihedral group with n even (viz., a • quadratic group), and herein, as we have just seen, lies the impossibility alluded to. § 9. Invariant Forms Belonging to a Group — The Set of Forms for the Cyclic and Dihedral Groups. True to the general process of thought which we have sketched in § 1 of this chapter, we now ask — after finding the homogeneous group of substitutions which correspond to the several groups of rotations — for all sa(Ai forms F{z^, z^ as remain unaltered, save as to a factor, for these substitutions. An invariant form (an expression which we shall hereafter INTRODUCTION OF x + iy. 51 retain) clearly represents, when equated to zero, a system of points on our sphere which remain unaltered for all rotations of the group m. question — a proposition which we can reverse. Now such a system of points must necessarily separate into mere groups of points of the kind which we have described in § 10 of the preceding chapter as appertaining to the group. The invariant forms which we seek therefore arise when any number of the forms which correspond to the aforesaid groups of points are multiplied together. Concerning the nature of the ground-forms thus presenting themselves, we can a priori make certain more detailed state- ments. If N is the number of rotations of a group, the groups of points which appertain to them consist in general of N separate points. The general ground-form will accordingly be a form of the N^ degree, and will contain besides — corre- sponding to the singly infinite number of groups of points mentioned before — an essential (not merely factorial) para- meter. But there occur among the general groups of points those in particular which contain only a smaller number of separate points. In accordance with this, special ground-forms, N of degree — will occur, which can only be considered as a special case of the general ground-form when we raise them to the I'*'' power. If we wish to push these general results any further, we must separate here the case of the cyclic groups from the others. In the case of the cyclic groups there occur among the general groups of points only two special ones, each consisting of only one point, viz., one of the two poles. Accordingly in their case there are two special ground-forms, and these linear on^. Retaiaing the system of co-ordinates which was introduced in § 4 in the treatment of the cyclic groups, these are simply 2 and z^ themselves. But further, we can here very easily construct the general ground-forms, and this by means of a method of reasoning which we shall find exceedingly useful in the following cases. To pass to the general ground-forms we construct the «"' powers of z^ and z^, and convince ourselves that, by the several substitutions (18), they acquire the factor ( _ 1)*. Whence we conclude that \z^ -\- \z^, understandiag by X : X an arbitrary parameter, is also an invariant form in 52 INTRODUCTION OF x + iy. each case. Since its degree is equal to n (equal to the number of rotations of the group), it is at the same time a ground-form. It is manifestly/, without further proof, the general ground-form. For we can so determine X^ : X^ that X^Zj" + X^z^" vanishes for an arbitrary point on the sphere, and therefore just represents the group of points proceeding from it by means of the rota- tions of the cyclic groups. Thus we have given a general solution, for the case of cyclic groups, of the questions which, first confronted us. We can express the result by saying that for the cyclic groups (18) the most general invariant form is given iy : (35) 2i« '^^ TI (^I'V + W) i where a, ^, denote any positive integral numbers and X^'\ X^''^ any parameters. In the other cases the theory presents certain differences, but only in so far as for them, among the general groups of iV separate points each, three of a smaller number of points occur. For the multiplicities which are to be attributed to these special cases, so far as we include them under the general ' groups of points, we will again assume the nota- tion v^, i/j, Vj, which we used in § 9 of the preceding chapter. N N N The said groups of points then contain respectively — , — , — , 'i '2 '3 separate points, and produce accordingly 3 special ground-forms ^j, F^, Fy respectively of the same degree. We construct F'^, F^-, F^\ Then it is shown that these powers all assume the same constant factor for the homogeneous substi- tutions in each case concerned. Therefore every linear com- bination : \F^^^+ x^^-^+^F,-., is an invariant form, and, indeed, as its degree shows, a ground-form. But the general ground-form contains, as we have said, only one essential parameter, while we here have two in X : X : X . We conclude that for the representation of all ground-forms it suffices to take into consideration the linear combinations : INTRODUCTION OF x + iy. 53 and tTiat therefore an identity (36) X WiF'j''. + Xj'oiif/z + XgWi^g^j = must exist between F^, F^, F^. Considering F^' always elimiaated by means of this identity, we have finally, as the expression of the most general invariant form: (37) i^i- . J-/ . F,^. JJi^^'-^F,', + x^'%^,), i where the positive integral numbers a, /3, y, and the para- meters X^'*\ Xj<'\ are throughout arbitrary. In the case of the dihedron, the whole theory here described presents itself again in such a simple form, in virtue of the position of the system of co-ordinates established in § 4, that we can write down the result immediately. We have : N=2n, ►i = »2 = 2, ►3 = n, and find accordingly : (88) F, = ^''",F, = '^,F,^z,z, F^ = represents the summits of the dihedron, F^ = the mid- edge points, i^3 = the pair of poles. Between F^, F^, F^, exists then in correspondence with (36) the identity (39) F^^-F^^-F^" = Q. As regards the tetrahedron, octahedron, and ikosahedron, the establishment of the special ground-forms requires in their case special considerations, to which we now turn.* * The forms Pi, F2, F3, considered in the several cases together with the rela- tions subsisting between them, occur for the first time in Herr Schwartz's memoir : " Ueber diejenigen Fiille, in denen die Gaussische Reihe P(a, /3, 7, x) eine algebraische Function ihres vierten Elementes ist," Borchardt's Journal, Bd. 75 (1872). See, too, frequent contributions in the Zuricher Vierteljahr- Bchrift from 1871 onwards. The reason of my only cursorily citing this funda- mental work is that its point of view in the treatment of the forms P is, in the first place, quite different from ours. Its starting-point is formed by certain qnestions in the theory of the conformable representation, on which we can only enter more fully in the following chapter. On the other hand, Herr Schwartz gives neither the groups of linear substitutions, nor the relation to the theory of invariants which we shall now lay so much stress on. 54 INTRODUCTION OF x + iy. § 10. Preparation for the Tetkahedral and Octahedral Forms. In the case of the tetrahedron and octahedron we have to distinguish, in accordance with § 5, two positions of the system of co-ordinates. Beginning with the first of these, we find for the summits of the octahedron (i.e., now the points of intersection of the co-ordinate axes with the sphere) the arguments : 2 = 0, c», ±1, ± i, and therefore the octahedron is simply given by the following equations : (40) z,z,{z,^-z,^) = Q. In a similar manner we determine the equations for the two corresponding tetrahedra and the cube determined by its 8 summits. The 8 summits of the cube have as co-ordinates We shall pick out the summits of one of the corresponding tetrahedra, if we choose here, among the 8 possible combina- tions of sign, those 4 for which the product ^»/^ is positive. Substituting in the formulae (6), we obtain for the arguments of the 4 summits of the tetrahedron : \+i \-i —\+i -\-i n/3-1 V3-1 VS+I s/3-l' Whence we obtain (by multiplying out the linear factors) the equation of the first^ tetrahedron in the form : (41) 2i* + 2 V~3 . Zj^/ + 2^4 = 0. . In the same way we find for the Cosniigr-tetrahedron (42) 2j4 - 2 s/^ . 2i V + ^2* = 0, and finally for the cube, on multiplying together the left sides of (41) and (42) : (43) 2j8+142iV+22* = 0- I will denote in the sequel the left sides of (40), (41), (42), (43), by t, $, ^, W. If we now rotate the system of INTRODUCTION OF x + iy. 55 co-ordinates, as we proposed at the end of § 5, through, an angle of 45° round the T-axis, these forms are transformed into others with only real co-eflBcients. I shall distinguish these forms by accents, and put : (44) )*' = 2i* + 2V3 . 2,V-^2S .W' = z^^-Uz^%i + z. 8 Equated to zero, these forms represent of course the octa- hedron, tetrahedron, and counter-tetrahedron, as well as the cube relatively to the new system of co-ordinates. § 11. The Set of Forms for the Tetrahedron. In accordance with the explanations given in § 9, our whole consideration of the tetrahedral forms may now be limited to two points ; first, to determine the constant factors to which the ground-forms : ( and '^ receive factors e ^ and e ^ , while t remains invariant for these also. The consequence is that, in addition to $' and '^^, $'4' = W also remains unaltered throughout, while $ and "^ themselves are only transformed into themselves by the substitutions of the quadratic group. As regards this latter circumstance, we perceive in it a confir- mation of a principle which we can establish a 'priori. This 56 INTRODUCTION OF x + iy. asserts that those substitutions of a homogeneous group, which leave altogether unaltered a corresponding invariant form, must form a self-conjugate sub-group within the main group of substi- tutions. These remarks, of course, just apply to the forms *', ^', t', w. Having confirmed by these remarks the existence of the supposed identity between $^, "4^^, t"^, &c.,* we shall be able to compute it by only taking into consideration the first terms in the expressions of ^^, '^^, t^. In this way we find without trouble : (46a) 12^"^ • i!^ -** + ■*'' = 0, or: (46b) 1 2 73 • ^2 - *'3 + y3' = 0. In connection with the results here obtained two remarks may be made which are both related to the invariant theory of binary forms, and of which the one m.ay express the signifi- cance which the said theory will often have for us in the sequel, while the other is designed to marshal the results obtained by us in the case of the tetrahedron relatively to the otherwise well-known products of the invariant theory. Suppose that, of the forms (45), we have only so far com- puted one, viz., $; then the theory of invariants supplies us with the means of deriving from it other tetrahedral forms by mere processes of differentiation. We have only to establish any covarianis of , equated to zero, represents the counter-tetrahedron, and similarly that the function-determi- nant, equated to zero, represents the corresponding octahedral form. For both these forms, equated to zero, must represent such groups of points as remain unaltered by the tetrahedral rotations, and no other groups of only 4 or only 6 connected points can exist besides those just mentioned, or at least do not come under consideration (inasmuch as the 4 summits of the original tetrahedron, which likewise form such a group, are already given by $ = 0). We should therefore he able to calculate also amongst the forms (45) ioth "^ and t hy constructing the Hessian form of $, and then, from this and $, the functional determinant. In fact, we get by calculating out directly : and : 32* «2* a^2 &1&2 «2* AS* ^2^h &2^ h &

: Hence the identity (46a) is included in the general relation (50) as a particular case, as was to be expected. We must, therefore, say that our geometrical reflexions on the group- theory have led us in the case of the tetrahedral forms not so much to new algebraical results, as to a new way to results otherwise known. § 12. The Set of Forms foe the Octahedron. Turning now to the octahedral forms, we already know, of the 3 special ground-forms appertaining to them, the two : and (t'=Z^Z^ (Zi* + 22*), (51b) tTF' = Zi8_l4ziV + 22*- We easily verify that, setting aside a numerical factor which occurs, W can also be computed as the Hessian of t. We obtain a new octahedral form by now constructing the * We arrive, of course, at the same result if we in general interpret geometri- cally on the sphere the double ratio of 4 complex values z—x + ii), in the way that Berr Wedekind has done in his inaugural dissertation (Erlangen, 1874), and in his note on the subject in the Mathematische Annalen (Bd. ix. 1875). INTRODUCTION OF x + iy. 59 functional-deteriniiiaiit of t and W. We thus hare, disre- garding a factor : (52) \X' = ^1^^ + 330iV - 33z,V - V'- We easily prove that this j^ is the third special ground- form of the octahedron, i.e., when equated to zero it represents the 12 mid-edge points of the octahedron. In fact, x~*^ must represent a group of only 12 point connected by means of the octahedral rotations, and since ^ is different from t"^ and the group of 6 octahedral point counted twice does not therefore come under consideration, there is, in fact, no other possible explanation. We have just seen that t and W remain entirely unaltered by the homogeneous tetrahedral substitutions. The same is consequently true of x- For ■)(_ being a covariant, can only alter by a power of the substitution-determinant at the most, if its ground-form is unaltered; but this determinant is in our case equal to 1. Now in § 1 we generated the homo- geneous octahedral substitutions by entertaining, in addition to the tetrahedral substitutions mentioned, a single substi- tution (23) which corresponded to a rotation Fof period 4. We determine by direct calculation that t changes its sign for this substitution (and therefore generally for all octahedral substitutions which are not at the same time tetrahedral substitutions). Accordingly W as the Hessian, and since we are again concerned with a substitution of determinant 1, remains generally unaltered, while ^ changes its sign alter- nately just like t, so that the product x< remains unaltered. In any case, <*, W^, x^> are in general not altered by our homogeneous octahedral substitutions, and there exists, there- fore, between them the supposed linear relation. Again, taking into consideration only certain terms in the explicit expressions which result for these forms from (51) and (52), we get for them : (53) \Qit*-W^ + x^ = Q, a relation which holds also for I/, W', and x'- The form t has been long known in the invariant theory of binary forms, inasmuch as it presented itself as the covariant 6o INTRODUCTION OF x + iy. of the 6tli degree of the binary form of the 4th order, when the latter was assumed to be of the canonical form : Similai'ly the synthetic geometers have on many occasions closely investigated the system of points t=Q, i.e., in their language : the aggregate of 3 mutually harmonic pairs of points. Clebsch, too, in his theory of binary algebraic forms, has considered the form < as a special case of general binary forms of the 6th order.* Finally, as regards the relation (53), this, with those analogous to it, are included under a general formula of the theory of invariants, by virtue of which the square of a functional determinant of two covariants is ex- pressed by integral functions of forms of a lower degree. § 13. The Set of Forms foe the Ikosahedron. To establish the form of the 12th degree, which, equated to zero, represents the 12 summits of the ikosahedron, we first calculate the arguments of the several summits, supported by our former developments (§ 6). One of the summits has the argument 2 = 0; introducing this into the 60 non-homogeneous ikosahedral substitutions (32), we obtain for the 12 summits : (54) Z = 0, 00, i" (£ + £*), £-(£2+e3) (►=0,1,2,3,4). We can therefore take the required form / equal to the follow- ing products : hh ■ JJ(h- '"{' + '*) ■ h) ■]~[(.h-'-'{'' + ^)^,l V V or : or finally : (55) /=Z,Z2(%^» +1121^-^2^")- We will now again calculate from the / so obtained, dis- carding the proper numerical factor, the Hessian form, and from this and / calculate the functional determinant. We thus obtain the two forms : * Cf. p. 447, &C. Consult too Brioschi, "Sulla equazione del ottaedro," Transunti della Accademia dei N. Lincei 3, iii. (1879), or Cayley, "Note on the Oktahedron Function," Quarterly Journal of Mathematics, t. xvi 1879. INTRODUCTION OF x + r. ly. 6i (56) E= 1 121 ay ay ay = - (2i«» + z^so) + 228 {z^^\^ - zH. iV=)- ■ 494zii%i'' (57) r= + 1 «/ 5^j «/ &, 20 3^ a., = (zjSO + g^30) + 522 (2^252^6 _ 2^62^26) _ 10005 (z^^^i" + z/Oz/o), and I assert with regard to them that 11=0 represents the 20 summits of the pentagon-dodeJcaJiedron, T=0 the 30 mid-edge points (the ends of the 15 cross-lines). In order to prove this somewhat more completely than was done in the analogous cases of the tetrahedron and octahedron, let us remark, first, that H and T as covariants of / certainly represent 20 and 30 points respectively on the sphere, such that their totality remains unaltered for the 60 ikosahedral substitutions. But now the points on the 2-sphere arrange themselves in general by virtue of these rotations iuto sets of 10, and the number of points thus grouped together is lowered then, and only then, and this to 12, 20, 30 respec- tively, when we have to do with the summits of ikosahedron, the pentagon-dodekahedron, and the mid-edge points. An aggregate of points which remains unaltered for the 60 ikosahedral substitutions must be a combination of such in- dividual groups of points. The number of points which it contains necessarily admits of being put into the form : a- 60 + y8. 12 + 7.20 + 3. 30, where a, /3, y, S, are integers, and /3, y, S, give the multi- plicities with which the summits of the ikosahedron, the pentagon-dodekahedron, and the mid-edge poiuts contribute to the aggregate of points. Now if, as in the case of H=0, this number is equal to 20, or if, as in the case of T=0, it is equal to 30, there is in either case only one possible determination of a, /3, y, S, viz., in the first case a = /3 = ^ = 0, 7=1, and in the second case a = (8 = 7 = 0, ^ = 1. But this is what we asserted re- garding the meaning of H= 0, T=0. 62 INTRODUCTION OF x + iy. We now investigate the behaviour of /, H, T, towards the homogeneous ikosahedral substitutions with reference to the factors that may occur. Considering only the generating substitutions (24), (26), we determine after a short calculation that / remains generally unaltered. The same, therefore, holds good for .2" and T. For we have defined IT and T as covariants of /, and the determinant of each substitution (27) is equal to unity. The behaviour of /, S, T, in this connection is thus as simple as possible. There exists, therefore, certainly, as was supposed above, a linear identity between/^, S^, T^. Again, recurring only to the initial terms of the explicit formulae (55), (56), (57), we find for this identity : (58) ^2= -.3-3 + 1728/5. We have thus found results which are quite analogous to those developed in the case of the tetrahedron and octahedron. If we are to demonstrate here also relations to the general theory of the invariants of binary forms, we cannot at any rate appeal to older works. For the knowledge of the forms /, ff, T, was, in fact, first obtained by the consideration of the regular solids and the circumscribed {x + iy)-sphere. I first investigated on this basis the principal invariantive properties of the form / in Bd. 9 of the Annalen (1. c). But there is a series of later publications on the theory of invariants. These are in connection with the definition, in the theory of invariants, of the form /, and of the other forms respec- tively, which we are considering. In this respect I had myself already announced in Bd. 9 of the Annalen the theorem that /, like the earlier forms $ and t, is characterised by the iden- tical evanescence of the fourth transvectant (/, / )*. This theorem Herr Wedekind had expanded in his " Habilitations- schrift," by showing that, apart from trivial exceptions, in general there is no other binary form whose fourth transvec- tant with respect to itself vanishes identically except $, t, and /.* Herr Fuchs has brought forward another property, ana- * "Studien im binaren Werthgebiet," Carlsruhe, 1876. See too, Brioschi, "Snpra una elasse di forme binarie," Annali di Matem., 2, viii. 1877. Latterly Brioschi has also considered such forms of the eighth order as are identical save as to d. factor with their fourth transvectant. See Comptes Kendus, t. 96 (1S83). INTRODUCTION OF x + iy. 63 logous to this, in his search for these forms,* viz., that all covariants of these forms which are of a loiver degree than the forms themselves, or are powers of forms of a loioer degree, must vanish identically. Herr Gordan then showed t that the pro- perty which underlies this is, in fact, just sufficient to charac- terise the form ^, t, f. I mention, finally, the latest work of M. H'alphen.'^ He starts, generally speaking, from the neces- sity for identities of 3 terms : and shows that these cannot occur otherwise than in the cases which we have investigated. We can thus even regard our forms as defined by these identities. These developments of M. Halphen are otherwise closely related to the others which we shall introduce in the fifth chapter of the present part, when our business is to establish generally all finite groups of binary homogeneous substitutions. § 14. The Fundamental Rational Functions. Having now spent sufficient time over the invariant forms which belong to the homogeneous substitution groups, it is easy to take the final step and construct such rational func- tions of s = - as remain in general unaltered by the non- homogeneous substitutions of § 7. In fact, we shall only have to establish proper quotients of our invariant forms of null dimensions in z and z^. We asserted in S 1 that in all cases one such quotient Z could be constructed, which, equated to a constant, uniquely represents in each case the different groups of points on the sphere such as we are considering. This is clearly nothing less than saying that there exists a rational function of the kind required which is of degree iV, under- standing by N the number of the non-homogeneous substi- * See the Gbttinger Naohtrichten of December 1875, as also the memoirs in Borohardt's Journal, Bd. 81, 8.') (1876-78). The " Primformen," which Herr Fuchs there considers, are just what we have called in the text " ground-forms." f Math. Ann,, Bd. xii. (1877) : "Bin. Formen mit versch. Covarianten." J "Mem. pr^sentds pav divers savants h, I'AcadfSraie," &c., t. 28 (1883) : " M^moire sur la reduction des dquations diff. lin. aux formes intdgrables (Prize-essay of the Paris Academy, 1880). 64 INTRODUCTION OF x + iy. tutions in question. Before we actually establish these fun- damental rational functions, and thus provide the shortest proof of their existence, it will be useful to make inquiries as to their position among the other rational functions which remain unaltered. I say first that every such rational function of z is a ra- tional function of Z. In fact, if R{z) be such a function, R{z) will assume the same value for all points on the sphere which proceed from it by means of the N rotations of the group in question, but the N points so connected are, by hypothesis, characterised by one value of Z. The functions Z and R, which, through the intervention of z, are always algebraical functions of one another, are therefore so related that to every value of Z only one value of R corresponds, i.e., ^ is a rational function of Z, q.e.d. That conversely every rational function of ^ is a function R{z), scarcely needs mentioning. I say further, tliat, hy the property attributed to it, Z is fully determined save as to linear transformations, viz., let Z' be a second rational function of z, which, like Z, has the property of representing, when equated to a constant, only one group of connected points. We conclude, just as before, that Z' de- pends rationally on Z, but that also Z depends rationally on Zi . Therefore Z' is a linear function of Z : Z! = -77 — -,- It is again manifest that we should be able conversely to use every Z' introduced in this way as our fundamental rational function just as well as the original Z. On the last remark is based the following : that we can subject our fundamental rational function Z to three more inde- pendent conditions, to make it fully determinate. First with regard to the cyclic groups, we simply put (59) Z=(|)" where Z therefore vanishes for one pole of the cyclic group, and becomes infinite for the other, and take along the equator the absolute measurement unity. In the other cases, we have always, as we know, to distinguish three special groups of points, which, with the multiplicities v^, v^, v^ respectively, are contained within the general groups of points appertaining INTRODUCTION OF x + iij. 65 thereto. Relying on a method frequently employed, we now so regulate our Z that it assumes for these three groups of points the values 1, 0, 00, respectively. Then Z will take the foi-m -V"^— J and Z—\ the analogous form — ' ^ 1 where by F^, F^, F^, are to be understood what we have previously called the ground-forms. At the same time c and c' must be of such a nature that the equation c . -J— - 1 = c . ^J— F^" J^s"' coincides with the oft-mentioned identity existing between -Fj, F^, F^, which fully determines c and c'. Turning now to the task of giving explicitly in every case the function Z thus defined, I make use of a notation which uniformly connects the two expressions of ^and Z—1, viz., I put Z : Z— 1 : 1 proportioned to We obtain in this form the following table, to which we shall often recur : (1.) Dihedron: (60) Z:Z-l:\ = ('-i^")': ('-1^'")'= " fe)" J (2.) Tetraliedron : (61a) Z:Z-1:1 = Y3: -12V33.<2:*3, or (61b) Z: Z- 1:1 = ^3: -12 V'^- !!'':*'', according as we assume the first or second position of the system of co-ordinates. (3.) Octahedron, with the same distinction : (62a) Z:Z-l:l=TP:x-:108^, or (62b) Z:Z-l:\ = W3:x'^:108t'*; (4.) Ikosahedron : (63) Z:Z-1 :l=a^:-r2:1728/^ 66 INTRODUCTION OF x + ly. For the symbols here applied consult throughout the principal formula of paragraphs 11, 12, and 13. § 15. Eemaeks on the Extended Groups. Finally, we return to our extended groups (| 7) once more. We want to know how our rational fundamental functions now obtained behave towards them. From the analytical side the extended groups 1. c. arose from a combination of the operation z' = z with the non-homogeneous groups of substitutions, where, so far as the tetrahedron was concerned, we only supposed the second position of the co-ordinate system to be employed. But now, maintaining the same supposition, all our ground-forms have real coefficients, and Z will be derived from these ground- forms, in virtue of the preceding formula, in every case by the help of real coefficients. The matter therefore simply comes to this : that for all those operations of the extended groups which are not already eontained in the corresponding non-homogeneous groups of substitutions, Z in each case passes over to its conjugate imaginary value. Combining this result with the propositions which we de- duced in § 11 of the preceding chapter, we obtain one final remarkable result. It is this : Z assumes real values for all those points of the z-spheix ^ohich lie in the planes of syminctry of tlie configuration in question, and only for stieh points. The points of the said planes of symmetry are therefore in each case characterised by the reality of the corresponding Z. Looking back, we have in the second chapter thus ended arrived at this point : we have connected the geometrical results of the group-theory occurring in the first chapter with a definite region of recent mathematic, namely, with the algebra of linear substitutions and the corresponding theory of invari- ants. Just in the same way, the following two chapters are destined to effect the connection with the two other modern disciplines. These are Riem.anns theory of functions and Galois theory of algebraical cqiMtions. ( 67 ; CHAPTER III. STATEMENT AND DISCUSSION OF THE FUNDA- MENTAL PROBLEM, ACCORDING TO THE THEORY OF FUNCTIONS. § 1. Definition of the Fundamental Pkoblem. The investigations of the preceding chapter have led us, in the formulae (59) — (63) of the last paragraph but one, to the knowledge of certain rational functions Z of s, which remain unaltered for the groups of non-homogeneous substitutions in each case considered, and by means of which all other rational functions of z, which remain unaltered, are expressed ration- ally. We annex to this result a statement of the prob- lems which we denote as the equation appertaining to the group in each case. We sttpposc, namely, that tlie numerical value of Z is arbitrarily given, and seek to calculate from it the corresponding z as the unknown; or, to express it differently: ■)ve no longer consider Z as a function of z, hut z as a function of Z. The equation which thus corresponds to the cyclic group is, according to formula (59), 1. c, none other than the binomial equation : The other equations correspond in just the same way to the formulae (60 — 63). I will collect them here briefly in the form : F "2 (2) -?l-a = ^' which we used incidentally in the preceding chapter. Here I' Fg, together with J^j, denote those three principal forms of which all other invariant forms are compounded as integral func- tions, and Vg, Vg are in each case taken from the table which was 68 THE FUNDAMENTAL PROBLEM. provided in § 9 of the preceding chapter, and which I reproduce here to facilitate reference : 1 1 "1 y-2 "3 N (3) , Dihedron . . . 9 2 n 2n Tetrahedron . 2 3 3 12 Octahedron . . 2 3 4 24 Ikosahedron . . ,1 2 3 5 60 I have here added a last column, headed by N, which marks the degree of the equation in each case under consideration.* But with the equations (1), (2), only a part of our earlier considerations is inverted ; we obtain a second mode of pre- senting the problem by recurring to the several invariant forms themselves. These forms remain unaltered by the homo- geneous substitutions of determinant 1 in general, save as to a factor. It is not difficult, however, to select from them these for which this factor is equal to 1, and which we can call the absolute invariants. The sequel shows that these absolute in- variants can be composed in every case as integral functions of 3 of them ; I have noted these three forms in the following table, together with the identities subsisting between them in each case : (4) I. Cyclic groii,ps. ( Forms : z^z^, z^s' I Identity : (ziZ^)'"'- ^2^" Z.,2". II. Dihedral groups. In the case of the dihedron we had : ^i 2 — 2 2 — ' 3~^l 2' and the relation F^^ = F^^ + F^\ * I shall also occasionally denote the degree of (1) by N in the following THE FUNDAMENTAL PROBLEM. 69 If we now seek the absolute invariants, we obtain for n even : (5a) l^°™«= F,\ F,\ F,F,F,; \ Identity : {F^F^F.f = F^^ . F^ . {F,^ - F,") ; and for n odd : fob) /Forms: F,\ F{-F„ F,F,; \ Identity : {F^F.^f . F^^ = (F^^F^) . F^F.^ - F./'+^l III. Tetrahedral group : * I Forms: F, = t, F^F,= W, }i^t>S; T"^ = & ^ I Identity: W^ = (b^ (^'^ - \2 J - 3 . t^). IV. Octahedral group : ,~. ( Forms : F.^ = W, F.^^ = t\ F^F^ = ^t ; '^ I Identity : (^O" = ^^ ( j^3 _ log^^). V. IJiosahcdral group : ,^ I Forms : F, = T,F,^ H, F, =/; ^ ' \ Identity : T"^ + H^ ~ 1728/5 = 0. We now suppose, in a particular case, that the numerical value of the three forms included in the table, in correspondence with the identity subsisting between them, is given, and we seek to calculate from this the values of the two variables z-^, z.^. Thus we have what we will call the form-problem. The number of the systems of solution of a form-problem is always 2N, where by JV^ is to be understood the degree of the corresponding equation. All these systems of solution proceed in this case, in just the same way, from any one of them in virtue of the 2N homogene- ous substitutions, as the N solutions of each equation manifestly do with respect to the N non-homogeneous substitutions. § 2. Reduction of the Form-Puoblem. As regards the solution of the form-problem, we can always accomplish it by means of the corresponding equation and an accessory square root. Take, for instance, the cyclic groups. We then calculate first from the forms (4) the right side of (1) : * In the case of the tetrahedron and octahedron, I now use, contrary to what 1 have hitherto done, non-accented letters. 70 THE FUNDAMENTAL PROBLEM. tben solve (1), whence we find -^=z, and finally obtain Sj, 2, themselves by introducing this value of — into the given form 2o of the second degree SjSo (which we shall now call X), whence : (9) ^2=y^.^l=^-^2- In the case of the other groups, the matter takes a form per- fectly analogous. For not only does the particular Z (2) in these cases also admit of being rationally composed of the forms (5) — (8), but we can also always construct rationally from these forms an expression which is of the second de- gree in Zj, Z.2. I choose as such, in all the cases : (10) -y= ^^- ^K If we have then determined, by means of (2), the quotients ^ = 2, we find, by comparison with (10) : (11) Z^^^/^ShlliL, 2,= ^ ^ ^ V ^(2,1) ' 31, 2, =Z. ■(2.1) where X (sj, g^) denotes the magnitude (10) previously given, and X («, 1) a definite rational function of z : F,{z,l).F^{z, 1) ^i(^, 1) ■ We have thus at the same time the means of simplifying the previous statement of our form-problem, of reducing it, as we will say.* By means of (9) and (11), Sj, Sj, depend only on X and Z, which, in their turn, are rational functions of the forms (4) — (8). We now introduce these values of Sj, %, into the forms (4) — (8). Thus these forms will be rational in X since they are all of even degree. But at the same time tliey will be also rational in Z. For they now represent rational function of z, such as do not alter for the N corresponding non-homo- * That such a reduction was possible was poioted out to me incidentally by Ilerr Niither, who derived it in a totally different manner from his researches on the representation of surfaces. THE FUNDAMENTAL PROBLEM. 71 geaeous substitutions. We shall therefore in the sequel, when speaking of the form-problems, not suppose, say, the forms (4) — (8) to be given [where we had always to pay regard to the identities subsisting between them], but rather the ex- pressions Z and X directly, and then consider %, z^, as functions of these two magnitudes. I reproduce here explicitly the rational functions of Z and X, to which the forms (4) — (8) are equal. "We verify these easily by reflecting, on the one hand, how Z and X are composed of the forms (4) — (8), and, on the other hand, taking account of the identities subsisting between these forms. I find : I. For the cyclic groups : (12) z,z., = X, z^- = Z . X'\ rj- = ^". II. For the dihedron : for n even : m + 2 (13,a) F.^- = , i'i^= :^ n+2 ^l-f^2^3= u Z^ and, for n odd : n + 3 (13b) ^,2 = ^^^, F,^F,^ U-^ ' Z ^ n+l i^l^2 = n-l Z 2 III. Fw the tetrahedron : X' .{Z-\f „p (14) F,-- 4^22 "^2^3- X^. {Z-\) 4322 ' ^2' = Z6.(Z-1)3 5184^^3. Z 72 THE FUNDAMENTAL PROBLEM. IV. For the octahedron : (15) F., = 10S.^^^, ^-108. ^%Z}1, F,Fo = 1082 . X^ ■ {^- 1)" V. For the ikosahcdron : (16) F, - 120 . ^-g-1'- ^,= _ 12a . £!!(|zi)!, ^3- ^ • § 3. Plan of the Following Investigations. We have now to discuss the fundamental problems, which we have thus far reached, under a double aspect, viz., in the sense of the theory of functions, and algebraically. Postponing the latter kind of investigations to the following chapter, we turn at once to the function-theory considerations. We have z, the unknown in the individual equation, as a function of Z alone, while the Sj, z^, of the corresponding form- problem depends also on X. But the mode of dependence by formulae (9) and (11) is so expressly simple that we need delay no longer over it. We will, therefore, only discuss z-^ and s, so far as they are functions of Z. Such an investigation breaks naturally into two parts. We have first to obtain a general survey of the different branches of our functions, and then to suggest the means of computing the particular branch of the function by a convergent process (for example, by a series of powers). We attain the former very simply, in our case, by the method of conformable repre- sentation (§§ 4, 5). We learn hereby, at the same time, the form of the series which come under consideration for the different branches of our functions (§ 5). The coefficients of the expansions will then be given by proving that z satisfies, in relation to Z, a simple differential equation of the third order, and consequently the roots z^, Zg. of l^ parallel form-problem appear as solutions of a homogeneous linear differential equation of the second order, with rational coefficients (^ 6-9). Finally, we THE FUNDAMENTAL PROBLEM. 73 prove in § 10 that, by reason of the last-mentioned differential equation, z^, z^ are particular cases of Eiemann's P-function, whereupon our investigations seem to adjoin a well-defined and much-explored region of modern analysis. As to the results which we obtain in this way, they are, in their main features, all contained already in the above-men- tioned work of Herr Schwarz ; * except that in Herr Schwarz's article the order of the matter is just the reverse of that fol- lowed by us here. Starting from the differential equation of the hypergeometric series, Herr Schwarz first constructs the differential equation of the third order, on which the quotient z of two particular solutions Zy, z^, depends. He then investi- gates the conformable representation which z projects from the two half planes of the independent variable Z, and ascends, finally, by means of the condition that 2; is to be an algehraical function of Z, to the z- functions considered by us and the fun- damental equations which define them.t We, on the contrary, begin with these equations, construct from them the conform- able representation, and then reveal the existence of the diffe- rential equations of the third order, which z satisfies, and, finally, pass from this to the differential equation of the second order of the P-function, or, what is essentially the same, of the hypergeometric series. In this connection it may be here ex- plained that, in taking this last step, we borrow an idea which Herr Fucks has introduced in his memoirs mentioned above,J inasmuch as we represent X{z.^, z^) (a form, therefore, dependent on 3j, Zg) directly by means of Z. I should, of course, have been able to collect the developments here described much more briefly had I desired to presuppose special knowledge with regard to Eiemann's P-function, or even merely to make use of the general foundations of the modern theory of linear differential equations with rational coefficients, as developed by Herr Fuchs in the 66th volume of Borchardt's * " Ueber dienigen Falle, in welchen die Gaussische hypergeometrische Reihe eine algebraisohe Function ihrea vierten Elementes darstellt." Borchardt's Journal, Bd. 75, p. 292-335 (1872). + I summarise in the text only such of the results obtained by Herr Schwarz as are in immediate relation with our own exposition. + "Zur theorie der linearen Differentialgleichung niit veranderlichen Coefifi- cienten " (1865). 74 THE FUNDAMENTAL PROBLEM. Journal. This sacrifice being made, my exposition acquires the importance of leading, by a relatively short route, to a portion of the researches just mentioned. I should like to refer here in this relation to § 3 of the fifth chapter following, where, in connection with the development now given, the most general linear difierential equations of the second order with rational coefficients, and which have entirely algebraical integrals, are directly determined. § 4. On the Conformable Eepeesextation by Means of THE Function z {Z). Turning now to the conformable representation which is furnished by z [Z), we denote as before the complex value of z = x + iy on the sphere, while we interpret Z=X+ iY gti a plane.* We construct in the plane ^the axis of real numbers, and decompose this into a positive and negative half-plane. We mark in addition, when we have to do with the binomial equations (1), the two points Z=0, c», in the other cases Z=l, 0, 00. A glance at the equations (1), (2), with reference to the more complete formulae (59) — (63) of the preceding chapter, teaches us that, in the case of the binomial equations, the n function branches coming under consideration for Z=0 and Z= 00 all congregate in cycle, while, in the other cases, for Z= 1, Vj^, of the If existing branches are connected cyclically ; for Z= 0, j/g ; and for Z = cc, v^. Now I say that the function z iZ) furnishes no other hranchings tJian those given here. In z general, viz., when Z is given as a rational function of z = — % in the form : ■4'(2i, Zj)' [where <^, i/r, are to be integral homogeneous " functions of the accompanying argument, of degree N^, we find those values of z, and therefore of Z, for which branchings take place, by * Whoever ife not thoroughly familiar with the theory of the conformable representation will consult with advantage Herr HolzmuUer's recently jmblished work, " Einfiihrung in die Theorie der isogonalen Verwandtschaft und der con- formen Abbildungen," &c (Leipzig, 1882). THE FUNDAMENTAL PROBLEM. 75 equating to zero the functional-determinant of the (2 i\^— 2)"" degree : ip d\j/ &^ ip &j " &!„ ~ 5zj ■ iz^ If this vanishes /i-times at a position z~z^, fi + l branches of the function z for Z= ^^ are connected cyclically in corre- spondence therewith * If we compute this functional-deter- minant in any one of our cases (1), (2), we always return to the branching points, which we already know. For in the case of the binomial equations we obtain simply : and in the case of the other equations, recalling that i^j is always — , and I\ is the functional-determinant of F^ and F, 3 ■ i^/'-'. F./z-'^ . F/=-^ = 0, where the different roots of Fj^ = all give Z= 1, those of i^j = 0, Z=0, and finally those of F^ = 0, Z = cc.f The data so attained are already sufiScient to characterise fully the nature of the conformable representation which we sought. If we describe as an w-gon every figure situated on the sphere, and furnished with the necessary number of sum- mits, and otherwise bounded by continuously curved lines, and observe that Z is rational in 2, and that therefore to every Z belong iV values of z, while to every z belongs only one value of Z, we have at once : In virtue of the binomial equation (1), the two half-planes Z will be alternately represented on 2JV lunes of the s-sphere which meet at the poles of the 2-sphere (i.e., the points s^z.^=Q) * The rule here formulated differs from that given in the text-books in the use of the homogeneous variables Zi, 22. This has the advantage of embracing in one form of expression the finite and infinite values of z, as the geometrical inter- pretation of z on the sphere and the modern conception generally of the infinite requires. f This explicit calculation of the functional-determinant was not really needed ; for the establishment of our result, it would have been sufficient to have remarked that the total number of the branching points for Z=^0, aa, and for Z = \, 0, oc, respectively (with their proper multiplicities taken into account) is identical with the degree (2iV- 2) of the functional-determinant. [We must here attribute (k - 1) roots of the functional-determinant in each case to v branches associated in cycle.] 76 THE FUNDAMENTAL PROBLEM. with angles = ^, and envelope the s-sphere completely, but no- where multiply. Just in the same way in the cases (2), the half-planes Z will be represented alternately on 2N triangles of the z-sphere, which, with angles equal to -. -j -, extend to one point of "i h 's J'j = 0, one point. of J'', = 0, and one point of i^3 = 0. We now observe that all roots of (1) or (2) are successively derived from any one of themselves, in each case, by N linear substitutions to which correspond rotations of the «- sphere round the centre. We thus conclude immediately that : The N lines or triangles which in an individual case corre- spond to the positive half-plane Z, as also the N lunes or tri- angles which correspond to the negative half-plane Z, are respectively congruent with one another. Finally, we recall the theorem which we deduced in the con- cluding paragraph of the preceding chapter from the existence of the extended group. We there showed that Z only assumes real values along those great circles of the 2-sphere which are traced out by the planes of symmetry of the several configura- tions. Now the real values of Z separate in the .^- plane the two half-planes. Hence we have finally : The boundary lines of the lunes and triangles are none other than the circles of symmetry before mentioned, and our lunes and triangles are therefore identical with those figures which we have described in § 11 of the first chapter as fundamental domains of the extended group. I beg the reader to make himself quite familiar with the formal relations here described ; this is not the place to discuss them more minutely.* The representation which corresponds to the binomial equations has of course been much investigated elsewhere, only that the z-sphere has been replaced throughout by the plane to which we must suppose our sphere related by means of stereographic projection. f * As regards the ikosahedral equation in particular, a glance at the figure gives the beautiful theorem : that this equaticm, for a real value of Z, jiossesses always four, hut only four, real roots. t In his "Vorlesungen iiber mathematische physik " (Leipzig, 1876), Heif Klrclwff describes those plane figures which correspond to our lunes as SicJuln. THE FUNDAMENTAL PROBLEM. 77 For the rest, I will in the developments of the following paragraphs leave on one side the binomial equations and the cyclic groups generally, in consideration of the gap which sepa- rates them from the other cases, and only note the simple results which relate to them in footnotes. § 5. March of the Zj, z^, Function in General— Develop- ment IN Series. The characteristic feature of the geometrical expression of the functions z [Z), as we have given it in the preceding paragraphs, consists in the fact that we have constructed, not a many-leaved surface on the ^- plane, but a region-partition on the s-sphere.* Having now to consider the march of the functions z.^{Z), z.^ (Z), we transfer our attention, accordingly, again to the ^-sphere. Leaving aside, as proposed, the cyclic groups, we have to recur to the formulae (11), which we will write in the following manner : (") "V^- i..M'/.'4'(,i) ' "•'■'^^ Here z^, z^, appear as single- valued functions of position on a two- leaved surface, covering the 2-sphere, which possesses branch- points at all points i^i = 0, or ^^ = 0, or #3 = (the point z = °^) ^^ we will write it : 77)^ -aji + 5^-^ = 0, consequently on differentiating successively with respect to Z 7 (rt'l + r,r) - an' + S^' = 0, 7 (ri'X + 2»!'r + iD - <"!" + K' = 0> 7 {n"'i + 3V'c + 3vr + 1'^") - «i"' + &r = In the three equations thus obtained, /3 has vanished of itself, the elimination of the other constants gives, after an easy re- duction : r r = InX 3VX' + 3i'r C" 1'" or, on separating the variables : r_3/rv='!"'_3/^"Y The differential expression required is therefore : We will in future denote this by [77] or by [■'7]^.* We will, moreover, here estimate how \j)\z varies if we introduce in- stead of ^ a new vaa-iable Z.^. If Z=F(Z,)Z' = ^,&c * According to a communication for which I am indebted to Herr Schioarz, this expression occurs in Lagrange's researches on conformable representation : " Sur la construction des cartes geographiques," Nouv. Mem. de I'Acad. de Berlin, 1779. Cf. further Herr Schwarz's often-mentioned treatise in Bd. 75 of Borchardt's Journal, where other literary notes are collected. In the " Sitzungs- berichten der sachsiscben Geaellschaft " of January 1883, I have tried to demon- strate what deeper meaning is involved in a differential equation of the third order [rj] —f(z) if we start from the origin of the expression [?;] as it is treated of in the text. THE FUNDAMENTAL PROBLEM. 8i there follow in order ; dZ^ dZ ' ■^' dZ-^ dZ-^ ^ ^ dZ' '^ ' dZ^ dZ^ ■^ ^ "^dZ^ ■ -^ ^ + cZZ ■ "^ • Therefore (23) Wz, = Wz--Z'^ + [^W which is the required formula. If, in particular, Z depends linearly on Z■^^, „_ AZ^+B CZ^ + D' then [Z]zj disappears, and we have simply § 7. CoNjfECTioN WITH Ldtear Differentul Equatioxs of THE Second Order. Before going further, ■2 - 2/2'Vi) +P(yi V2 - ViVi) = 0- We have further : (28) ih]hz^=r,; whence by logarithmic differentiation : Vi'Vi - yi'y\ _ 2 ^' = ^' y^Vi-yivi 2/2 "" or, by virtue of (27) : (29) X'=-i>-2^'. On further differentiation it follows that : 1X1/ 2/5 V2/9/ n \n / - y^ \ 2/2 and therefore, by combination with (29) : Now the terms which here, on the right side of the equation, contain y^ are just equal to 2g by the differential equation of the second order to which y^ is subject. We therefore find : (30) [')]z = 22-lp2_y, which is the final formula which we sought. If to every linear differential equation of the second order (25) there thus belongs a definite differential equation of the THE FUNDAMENTAL PROBLEM. 83 third order (26), then clearly to every dijQFerential equation (26) belong infinitely many equations (25). We have only to put (31) ^-\p^-p' = r, and in this, p (as a rational function of Z, if we lay stress on that point) can still be taken arbitrarily, q being hereupon uniquely determined (and in fact again as a rational function oi Z Up and r are rational). Evidently (26) is completely solved, if one of the correspond- ing equations is so too. Conversely, too, the solutions of (25) are very readily given if the solutions of the corresponding equation (26) is regarded as kno^vn. We conclude, namely, from (27) by integration in the well-known manner : (32) 2/a'2/2- 2/2^1 = ^^^'^, understanding by k the constant of integration. Combining this with (28), there results : (33) { 1^ -Ifpdz 2' 12/2--" ~- ■ * The linear differential equation of the second order, therefore, requires, afber previous solution of the corresponding differential equation of the third order, only a single ciquBiJ s r aa t besides in order to solve it. ff . _# -A- § 8. Actual Establishment of the Differential Equation OF THE Third Oedee for z\_Z'\. In order now to actually establish the differential equation of the third order : Wz = '-(^. which our z satisfies as a particular solution, we make use of what is contained in formula (19) with regard to the develop- ment of («— Zg) in a series according to powers of [Z—Z^. We consider the developments in series to be explicitly written down, and from them a series calculated for [2]^ by direct differentiation. As initial term of this series (which, besides, 84 THE FUNDAMENTAL PROBLEM. must proceed according to integral powers of (Z—Z^) since [z]^ is a rational function of (Z), we have for Z^ = 1, 0, oo, respec- tively : 1^ 3-1 ^2^ -1 3^^1 Now I say further, that [z]^ will certainly not become infinite for a position Z^ which is different from 1, 0, or ex. At such a position we have, viz. (as follows again from the conformable representation) : z-z,==a{Z-Z,) + h{Z-Z,y+. . . where « < 0, and hence for [s]^ a series proceeding by integral powers of (Z—Z^) and only possessing positive exponents. We put in accordance with these results : where A, B, C, will be constants, and these we must now bo determine, that the development in series, which r (Z) admits in ascending powers of -^ in the neighbourhood o{ Z = co, shall ... . . v2_ 1 possess the initial term just given -„^„-™,. The, result shows tJuxt A, B, C, are completely determined hy this necessity. In fact, we have immediately : Vl^ - 1 k"" - 1 . . ►,2_1 Introducing these, our differential equation will lie simply : 1 + 1-1-1 ^ ^ '- -"^ 2.i2(z-l)2 2>^^.Z^ 2 {Z-1)Z ' where now for v-^, v^, v^, the numerical values of our table (3) may be substituted.* The three critical points Z= 1, 0, oo, just because one of them lies at .2" = oo, do not enter into a differential equation * For the binomial equation (1) we get as the corresponding difEerential equa- tion by direct diSerentiation : -1 1 C=0,A^B = Q,^^'^^A = ^^ Wz 2n2 ■ Z'^ THE FUNDAMENTAL PROBLEM. 85 with a symmetry corresponding to their peculiar importamce. We shall at once remedy this if we introduce in place of Z as a new variable some linear function of Z, which for Z=l, 0, 00, assumes any three finite values a^, a^, a^. Making use of the formula (24), and, further, calling the new variable itself Z again, we have : 1 r 2 1 where now, as we see, all desirable symmetry reigns. § 9. LixEAE Differential Equations of the Second Order FOR «i AND Zg. The developments of § 7 put us in a position to give the most general linear differential equation of the second order with rational coefficients : (36) y"+p.7/ + q.y = 0, which has two particular solutions y^, y,, whose quotient is equal to our z; we have only to put according to formulae (31), (34) : 1+1-1-1 ^ 2^ -^ 2.i2(Z-l)2 2v^-^.Z-i 2{Z-l)Z I say now that amoiig these differential equations there is always one which the roots Zj, z^, of our form-prohlem satisfy. In fact, we recognise a priori that Zj, z^, must be particular solutions of a linear differential equation of the second order with rational coefficients. Namely, let z-^, s.°, be two corresponding branches of our functions, then any other branches express themselves as linear homogeneous functions of these Sj", z^. They there- fore all satisfy the following differential equation : • y y y dZ-^ dZ 1 dZ^ dZ ' 86 THE FUNDAMENTAL PROBLEM. We now conclude at once that the coefficients, which y", y, y, obtain when this determinant is developed, behave as rational functions of Z. They are themselves, indeed, without further consideration rational functions. For if we replace z^, z^", by any other pair of corresponding branches of 2j, z^ : these coefficients, since aS — /37 by virtue of the definition of the form-problem = 1, remain altogether unaltered, according to the rule for the multiplication of determinants. Our next object is to seek, out of the totality of the differential equations (36), the one which z^ and z^ satisfy. Let 2/1, 2/2' ^® *^° solutions of (36), such that -^ = 2. Then vie, Hi will first calculate generally A (2/. 2/,) F,ljj;,y,) To this end we start from the equation ' ^3-' (2. 1) Differentiating this, and considering as before that F^^ is always, save as to a numerical factor, the functional determinate of F„ and F^, we obtain (c representing a proper constant) : , Fi^{zjyj\{z^) F,^,^\z,\) "•' '' or, on introducing another appropriate multiplier c" : c" . z . -^j:i^i2) ..'=1. F^{z,\).F^{z,\) Here let us now put z=-^. Then c" .z ^1 ^yv_ yJ (v 'v -y'v)-i ^Avv y«:)'F,{y„ y,y^y^y^ y^y^^-^' or finally, embodying the symbol X and the formula (32) also : (37) X{y^,y^) = h.c" .Z.e-^^', which is the formula we required. THE FUNDAMENTAL PROBLEM. 87 Now for the solutions z^, z^, of our form-problem, not only was Z-, -=2, but it was determined tbat X {z.^^, z^ was to be inde- pendent of Z. We shall therefore have to take the coefficients p of the corresponding linear differential equation in such a way that Z in general disappears from the corresponding formula (37). This gives, as we see, ^ Z Introducing this value into (36), we obtain the diflferential equation which we sought. This, after some easy modifications, runs as follows : § 10. Relations to Eiemann's P-Function. We now have all we require in order to calculate by a series of powers s^, z^, and from them z= J, in the neighbourhood of any position Z= Z^. In fact, we saw in | 5 how we could deter- mine in an individual case the nature of this series of powers, and have now simply to substitute the series itself in (38) in order to find the coefficients in the series which still remain unknown. If we wish to effect this in particular for the neigh- hood of the point Z = za, we can use the formulas (21) im- mediately. If I do not more explicitly carry out the step here proposed, nor discuss more closely the convergence and' the analytical law of progression of the developments suggested, it is because we have meanwhile obtained all the preliminary conditions for basing the investigation of the functions Zj, s,, on a ready-pre- pared and well-known theory. I mean the theory of Eiemann's P-functions : * For the solutions Zi, 22, of the form-problem of the cyclic group, we find in a similar way : y"+t ^_ = 0. ^ Z iri-Z- 88 THE FUNDAMENTAL PROBLEM. and the representation of their several hranclies hy the hypergeo- metrical series of Gauss* I tave already said that I will not take for granted any previous special knowledge concerning the P-fanctions. We may therefore define these functions in the way which most conveniently fits in with our previous develop- ments, viz., as solutions of the following differential equation of the second order : _ [aa' - (aa' + p^' - yy') x + PfS'x^ = 0, x^l-xy^ where a + a' + ^ + /3' + ^1 = ^1 W •••-5^-1= ^'^-i (-Ro)- G 98 THE ALGEBRAICAL CHARACTER OF Here the i/r/s denote rational functions of the accompanying argument, which are only so far completely determinate that we will not modify it by the help of the Galois resolvent itself, and i/^oC-^o) ^^ '^^ course only written instead of Sg itself for the sake of uniformity. We select one of these formulas and write (neglecting the former indices of the iB's) : (3) i^'=^^<(i^), and consider the Galois resolvent transformed by the help of this formula (by eliminating the B between the resolvent and the formula (3). Thus arises an equation of the degree W for B' which, in any case, has the root Hi in common with the original Galois resolvent. Now, the resolvent is by hypothesis irreducible. Hence the two equations of the iV^* degree have all their roots common, i.e., they are identical. We have therefore the theorem : The Galois resolvent will be transformed into itself iy the N rational transformations (3). If we therefore substitute in formula (3), instead of R, any root iZj, R' will become equal to another root Rj. But, instead of Rk, we can write •>|rj(Rg), and ■>]rj{R^ instead of J?j. Hence : ^^,(i^„) = ^i.,^^,(i^„), and therefore generally : so far, namely, as we disregard the changes which can be wrought on the individual symbols of this expression by the help of the Galois equation satisfied by the Ri. In this sense we have : Tlie N rational transformations (3) form a group. We ask how this group is connected with the Galois group [~. If we replace, in the formula (2), the R^ on the right hand by Rg, Rj^, . . . Rx—i in order, we obtain on the left-hand side, in consequence of what has just been said, the roots J?^ again, in each case in unaltered sequence. We obtain, therefore, iV" different arrangements of the R's, and now the assertion may be proved that those N permutations, hy which these arrange- ments proceed from the original arrangement, just make up the THE FUNDAMENTAL PROBLEM. 99 group p. For this purpose we will show that a rational func- tion of the Ela F{R^, Ry . . . Ri,^-^, which remains unaltered when we replace the sequence M^^, iJj . . . ^jv-— 1 I'y *^y of the other N orders in question, is rationally known. In fact, every rational function of the i?/s can in virtue of (2) be compressed into the form $(i?o). If, now, F submits to the changes mentioned, it will be just as truly equal to ^{R^, or equal to ^{R^, &c., understanding in every case by

Kr):x('-) = ^:Z-l :1, * I gave this, in the form here used, first in Bd. xii. of the Math. Ann. (1877), p. 517, &c. + See "Annali di Matematioa," Ser. I. t i., 1858. It; I first communicated the principal resolvent, in a somewhat less simple form, however, in Bd. xii. of the Annalen, p. 525. It is also implicitly the foun- dation of the parallel investigations of (Jordan, which we shall only describe in detail in the following Part (see in particular Bd. xiii. of the Annalen, " Ueber die Auflosung der Gleichungen 5 Grades," 1878). THE FUNDAMENTAL PROBLEM. 109 where ^, •, -^fr, ■)(. It is clear, in the first place, that the aggregate of the points /=0 will be permuted amongst themselves for the 12 rotations which leave r unaltered (i.e., for the 12 rotations of the cor- responding tetrahedral group). Therefore r will assume the same value for all points of /=0. Hence ■xir) is necessarily the fifth power of a linear expression. We consider further the 30 points T= 0. Amongst these are found, above all, the 6 sum- mits of the octahedron belonging to the tetrahedral group (which we just now denoted by t). The remaining 24 points are divided (as is evident on a model) in virtue of the tetra- hedral rotations into twice 12 associated ones. We hence conclude that "^(r) contains one simply linear factor, and two otJiers counted twice. Aa regards these multiplicities, let us remark that i/r(r) = 0, corresponding to the term T^ = {z) of the ikosahedral equation, must represent the aggregate of points under consideration, counted twice. The linear factor, however, which vanishes at the 6 octahedral summits, will be of itself twice equal to zero ; it need therefore be only counted once as contained in ■{r) = or H = 0. Among them are found, as we know beforehand, the 8 summits of the cube W appertaining to the tetrahedral group. These distribute themselves in virtue of the tetrahedral group into twice 4 co- ordinated points, of which each remains fixed for 3 tetrahedral rotations. We have, in addition to these, 12 more points of H= which in respect to the 12 tetrahedral rotations form a c (r- -af(r^- -Pr + y) c' (»•- -S) (r^. -v+lf c" (r- -1)^ no Ti/£ ALGEBRAICAL CHARACTER OF single group. Hence we conclude that ^(r) possesses only 3 different linear factors, of which the two which correspond to W= occur simply, while the third occurs as a cube. Summing up, we have reached a resiilt which expresses itself when we replace the formulae (17) by the following: Z:Z-l:l (18) understanding by a, /S, 7, . . ., c, c, c," constants which are still unknown. The determination of these constants is a problem which is only determinate when we have previously defined r in an unambiguous manner. Let r be one of the triply infinite number of rational functions of the twelfth degree, which remain unaltered for the rotations of the tetrahedral group. We will now put, in particular, r = -, understanding by t (as above) the octahedral form appertaining to the tetrahedral group. Here t should be so chosen that, when arranged in powers of g^, Sg, it begins with the term +Zj^ and has altogether real coefficients.* Then the first result is that, in (18), c" (r—jf)^ is equal to C (since it is only to vanish for r = 00), and therefore c is to be put = c' while S vanishes. We have further that C is to be taken =— 1728c;. For -' in consequence of our convention, reduces itself, for a very large value of — , to '2 -' as a first approximation, while Z (in virtue of the ikosa- '2 -2 B hedron) is to be replaced by j* y Finally, it follows that all the coefficients in (18) will be real. We have therefore now so simplified formula (18) that we can write : * Both these conditions can be satisfied, as a glance at the figure shows. For on the one hand, each of the 5 octahedra occurring in connection with the ikosa- hedron contains a temi with zi', because none has a summit at z-^ = 0, and, on the other band, amongst these octahedra is found one which has the meridian of real numbers for its circle of symmetry. THE FUNDAMENTAL PROBLEM. iii 2:Z-l:l = (r-a)3(r2-/3r + y) (19) -.rir'^-ir + r^f : - 1728, understanding by a, j8, 7, e, f, real constants. Now a, /3, 7, 6, f, must in any case, in correspondence with this formula, be so determined that the following relation is identically true : (20) {r - ay (r^ - /3r + 7) + 1728 = r {r^-er+ Q2. On treating this identity by appropriate means, we recognise that, with its help, a, /8, 7, e, f, are fully determined. Namely, we have first, on putting in (20) r = : a?y= +1728. Then, on difiFerentiating (20) with respect to r, we find further : (r2 - af (o)'2 - (2a + 4^) r + (0^8 + 87)) = (r2-£r+0 {5r^-3er + Q, or, since (r^ — er+J) and {r—ay are necessarily prime to one one another : 5e = 2a + ip, 10a = 3£, 5^= a/3 + 37, 5a2 = ^, therefore (by eliminating e, f) : lla=3y8, 64a2 = 97, and by combination with the relation first found : a5 = 35. But now a is to be real. Thus we have a = 3, and hence yS=ll, 7 = 64, e=10, f=45. The resolvent of the r runs therefore simply thus : Z:Z-l:l = (r-3)3 (r2_llr+64) (21) :r (?-2-10r + 45)2 : -1728. § 10. Computation of the Forms t and W. We now give a supplementary computation of the forms t and JF whereby on the one hand, we attain to an explicit exposi- tion of the connection between the quantity r, used in the pre- ceding paragraph, and the ^ of the ikosahedral equation, and, 112 THE ALGEBRAICAL CHARACTER OF on the other hand, obtain the necessary foundation for the invariant-theory method. "We remarked in § 12 of the first chapter, that to that tetrahedral group which we have here to consider belong the rotations : T, U, TU, to which we then made to correspond, in § 7 of the second chapter, the substitutions : 2 :r; (^- -^) z + {e^- -.3) 1 -^) .-(e*- -)' z' = z' , (.2. -^1 2+ (e- -^) z = c^- -*) 2- (.2. -e3)- We compute, in a homogeneous form, for the pairs of points which remain fixed for these substitutions the following equa- tions : 2^2_2(.2 + e3)2jZ2-z/=0, Z^'^ + Z,^ = 0, Sut noio the octahedron t will he constructed with just these 3 pairs of points. On further reflecting that the form t must contain the term -l-^^^ we have, accordingly, for the latter: t {Z„ Z,) = (2,2 + ^2) . (,^2 _ 2 (5 + e*) 2,2, - 2/) (22) .(z,^-2{e^ + ^)z^z,-z,^) = ZjO + 22,^2^ - 52/2/ - 52i2z/ - 22,2/ + z/. If we now wish to compute the corresponding W, this can be done, according to our earlier developments, on establishing the Hessian form of t{z^, z^). We may further agi-ee, as it is convenient for our later calculation to do, that W(z,, 2„) is to contain the term —z^- We have thus: W (2,, 22) = - Zi^ + 2/22 - 72, V - 72, V -H72,32/^72,V-^1^/-2/, (^^) 4.7,3^5 and we have thus already achieved the first object of the present paragraph. THE FUNDAMENTAL PROBLEM. 113 We now subject t and W to the operations : Thus arise, respectively, those five values which always come under consideration simultaneously in the case of our equations of the fifth degree, and which we will call t^, W^. We find : IF. (Z„ 2^) = - ^'Z,^ + ^■'Z,\ - 7 €2.Zj0z/ - 7 e-2,%3 + 7 ,4.^^3,^5 ^■^ '' - 7 eS-ZiV - £2-ZjZ/ - £-z/. Here we will inquire expressly how the five ^^'s or WJs are permuted under the 120 homogeneous ikosahedral substitutions. This, however, is derived already from the statement which we have made in § 8 of the first chapter concerning the correspond- ing geometrical figures ; but it seems useful to blend the rule in question explicitly with our present formulae. We have generated the 120 homogeneous ikosahedral substitutions from the following formulae by repetition and combination : S: 2{ = ±i\, z^ = ±i\, T: + V5.z;=-(£-€*)Zj + (£2_eB)z,, Introducing now these values of z.^, z^, instead of 2j, 2,, in the forms ty (or the W^, new forms t'y arise, whose connection with the original tjs is given, after a little calculation, by : ^^°^ \T:t^ = t„ t,' = t-, t-; = t„ 1^ = t,, tl = <3. Here, in the formula for ' ^>) ^ill sXso, ia virtue of the same considerations, be identically zero. Hence, generally, to our equations of the fifth degree those ones will belong whose roots are linear com- binations of the TTy's and the t^WJs with constant coefficients : (30) Ty=IS -W^+Tt^W,. We accordingly set ourselves the task of calculating out, for any values of o-, t, the corresponding equation of the fifth degree. Inasmuch as the details of the calculation offer nothing of special interest, I communicate the result imme- diately. We find 75 + 5^2(8/2. ff3+r-ff2l- +72/3. or2+^y ^) (31) +5F(-///- ff* + 18/2/r. e"-T-i + nT- ar^ + 27pH. r*) + {H' ■ ffS - 10///2 . ffS^S ^ 45/2//2 . „^i + y^2 . y6) = 0. In order to construct herefrom a resolvent of the ikosahedral equation, we have but to recur to the formula (28), and put instead : (32) ^=^r-- Then we can write in consequence of formula (30) : (33) Y^ = m-v^ + n- u^v„ ^ s-H r-HT where m is put = ^j^ , n = yj|^. On introducing into (31) the values of a, r, resulting herefrom, we obtain rr TT^ CTT-o/d Q 1 fl 9 6m?i2 + 7j'\ Z-T^+ 5r2 (8m3 + 12TO2/( + _- — -— - ) \ (1 -Z) / (34) -t-loF f -4m*. {\-Z) ^i-zr 0. This is that resolvent of the fifth degree of the ikosahedral equation which we shall later on denote as the principal resolvent. ii6 THE ALGEBRAICAL CHARACTER OF § 13. Connection of the New Resolvent with the Eesolyent of the r's. We have now to exhibit the connection of our new resolvent with the resolvent of the r's (§ 10). First, as regards the agreement of the function-theory and invariant-theory methods, we write the equation (27), say, as follows : (35) T= !!(<*- 10/^2 + 45/2); now squaring, dividing both sides by /^, and finally writing r again for — , we have : - 1728 {^- 1) = r(r^ - lOr + ibf, an equation which, in fact, is identical with (21). We shall have further to express 12<./2 , 12PF./ u = — y^ and v=^^ rationally in terms of r. As regards u, we achieve this at once on introducing for T the value (35). We thus find : To exhibit v similarly, let us recall that, according to the de- velopments of § 10, the points t^=0 are at the same time TT represented by ?•— 3 = 0. Therefore ^fr w^iH he identical with t^—Zf, save as to a factor. The comparison of any term in the development in terms of z^, z^, shows that this factor =+1. Hence we have without further proof: Finally, introducing the values of (36), (37), we have: \%fa ( r-3)-144w ^ ■ ' (r-3)>3-10r + 45)' THE FUNDAMENTAL PROBLEM. 117 We should, of course, be now able to compute also the resolvent of the m's and the principal resolvent, on eliminating r between (21) and (36) respectively* § 14. On the PfiODucTs of Differences for the u'a AND THE y's. We now further calculate, also in view of their later appli- cations, the products of diflFerences of the m's and the F's, which, as we know, are rational in Z. We consider, say, first, the following product : Hit. ■tA where the symbol under the product-sign is to denote that only those 10 factors are to be multiplied out for which v is , = {ez,^ + 2z,z,-^''z.?f. * Compare Hath. Annaleii again, Bd. xii. pp. 517, 518. I20 THE ALGEBRAICAL CHARACTER OF Now let the equation of the sixth degree, which the ^'s satisfy, be : <^8 + a>5 + 6'<^4 + c'(p + d'4>^ + e'4> +/' = 0, then a, h', c . . . are ikosahedral forms of the 4th, 8th, 12th . . . de- grees respectively. Hence it follows at once that a'=b' = d' = 0, while e, e, /', must, apart from numerical factors, coincide with /, II, and /^ respectively. We determine these factors in the well-known manner by returning to the values of/, ff, and (f> in 2j, z.^. We thus find with little trouble the following equation : (43) 4>^-\0f.4>^ + H.^ + 5p=^Q. Let us now concern ourselves for a moment with the group of this equation. This will be given, as follows from our earlier developments, by those 60 permutations of the „ remains unaltered for S, while ^„ is transformed into ^„+i. We can compress this into the single formula : »' ^= V + 1 (mod. 5), inasmuch as for v = 'x> the v so determined will also be oo. On the other hand, for T, H -<^6 + 10/.<^3-5/2 and therefore : (50) ? 12/2 12/2 $2 + 10f-.5 12 § 16. Concluding Kemarks. The developments of the last paragraphs have manifold rela- tions with the applications which are going to be made of them in the part here following. It has thus been already noted that the considerations of the present chapter will be of the weightiest importance for our further process of thought. Let me state this more precisely. We have already seen, in the third chapter of the present Part, that we can consider the solution of our fundamental equa- tion, from a function-theory point of view, as a generalisation of the elementary problem ; to extract the w**" root from a magnitude Z. The algebraical reflexions of the present chap- ter have then shown us that the irrationalities which are introduced by the equations of the dihedron, tetrahedron, and octahedron can be computed by repeated extractions of roots. The ikosahedral irrationality, on the contrary, has maintained its individual importance. Hence an extension of the ordinary theory of equations seems to be indicated. In the latter we are generally restricted to the investigation of those problems which Compare Mathematische Annalen, xiv. p. 143 (formula (19) of that page). THE FUNDAMENTAL PROBLEM. 123 admit of solution by repeated extraction of roots. We will now adjoin, as a further possible operation, the solution of the ikosa- hedral equation, and ask whether, among the problems which do not admit of solution by mere extraction of roots, there may not be some for which this can be effected by the help of the ikosahedral irrationality. In this sense our second Part now deals with the general problem of the solution of equations of the fifth degree. The attempt to accomplish this solution with the help of the ikosa- hedral equation appears the more natural inasmuch as the group of the equations of the fifth degree, after adjunction of the square root of the discriminant, is holohedrically isomorphous with the group of the ikosahedral equation, and as we have, in the resolvent of the fifth degree of the ikosahedral equation (previously established), the same number of special equations of the fifth degree, whose relation to the ikosahedral equation is a priori fixed. ( 124 ) CHAPTER V. GENERAL THEOREMS AND SURVEY OF THE SUByECT. § 1. Estimation of our Process of Thought so far, and Generalisations thereof. Having now, in the third and fourth chapters, studied the essential properties of our fundamental problem, we will inquire where lies the proximate cause of the remarkable simplicity which has manifested itself therein all along. About this, I believe, there can be no doubt, viz., it is the property/ of our proUems that froin one, of their solutions the others ahvays proceed 1)1/ means of linear substitutions which are a priori known. The geometrical apparatus, from which we started in the develop- ments of the first and second chapter, has served to lead up to our problems, and to illustrate their primary properties ; now it has done us this service, we can henceforward leave it on one side.* Forming this conception, we shall naturally ask if there may not exist other equations, or systems of equations, also which agree in that most essential point with our fundamental problem. We therefore first seek, so far as it is possible, for new finite groups of linear substitutions of a variable z (or two homo- geneous variables z^, z^. But we will show immediately (§ 2) that all such groups return to the ones already known to us. If we, therefore, conceive our statement of the question in the obvious manner explained, the equations and systems of equations * This is only meant to apply ad hoc, and for the developments of the second Part here following. For carrying out more thoroughly the generalisations pro- posed in the text, an illustrative notation, at all events when we have to deal with transcendental functions, is for the time quite indispensable, as also in § 6 of the present chapter, where we involuntarily, so to say, return to geometrical explanations. GENERAL THEOREMS, ETC. 125 hitherto treated of are the only ones of their kind. This is a result which is calculated to attach a certain absolute value to our previous considerations, which, on account of their induc- tive form, at first appear to aim at no definite object. In fact, we see that our fundamental equations occur as a specially remarkable circumscribed group among numerous mathematical investigations of the last few years. In regard to this, I will bring forward, in § 3 following, the simple developments by means of which we show that, with the help of our fundamental equations, all linear homogcTieous differential equations of the second order with rational coefficients, ivhich have entirely alge- braical integrals, can be established with little trouble. I refer, however, for the analogous significance of our fundamental equa- tions for the linear homogeneous difierential equations of the n*^ order with rational coefficients, to the memoir of Halphen * already quoted ; further, as concerns the role which our funda- mental equations play in the theory of elliptic modular func- tions, and similarly in the investigation, by the theory of numbers, of binary quadratic forms, to my own investigations -f- and those of Herr Gierster.j Meanwhile we can generalise our statement of the question in a twofold sense. In the first place, we can, instead of the variables ^j, z^, take into consideration a larger number of homogeneous variables, Zj, Zg) ■ • • ^1" ^^^ inquire for the finite groups of linear substi- tutions which may exist in their case. I will presently (in § 4, 5) treat this more fully, and will here only observe that, as a consequence of the views thus unfolding themselves, the developments of the second Part here following appear as a single contribution to a general theory, which embraces the whole theory of equations. Our second generalisation proceeds in another direction: we wUl retain the one variable s = '\ but, on the other hand, take into consideration infinite groups of linear substitu- * "Sur la reduction des equations diff^rentielles lineaires aux formes inte- grables." M^moires presents, &c., xxviii. 1 (1880-83). t Cf. especially Bd. xiv. of the Math. Ann., p. 148-160 (1S78). J " Ueber Relationen zwischen Classenzahlen binarer quadratischer Formen von negativer Determinante," Erste Note (Gottinger Nachrichten of June 4, 1879, or Math. Ann., Bd. xvii. p. 71, fee). 126 GENERAL THEOREMS AND tions. Here that vast region opens out, single-valued transcen- dant functions, with linear transformations into themselves, to which general attention has recently been drawn from various quarters, but particularly by M. Poincar^.* It is, of course, impossible for me to enter more minutely into the questions connected with this matter in the following paragraphs. My exposition is only to carry us so far that the position of the simplest class of functions among the others, viz., the elliptic modular functions, may be clearly conceived. To this is attached the proof (§ 7, 8) that the equations of the tetrahedron, octa- hedron, and ikosahedron admit of solution in a similar manner to that in which, say, a binomial equation is solved by logarithms, a cubic equation (and also the general equation of the dihedron) by trigonometric functions ; and this proof I wished to bring forward in its general outline, because it describes that point on which in the theory of equations, and particularly of equations of the fifth degree, the interest of mathematicians has been continuously concentrated. We can, evidently, also combine the generalisations here suggested ; we can study transcendental functions of several variables with an infinite number of linear transformations into themselves.f But more important for us here are, I think, the considerations which I develop in § 9, in consequence of which absolutely no material difierence exists between the two kinds of generalisation. Hence the per- spective to which the consideration in § 5 of the finite groups has already led us will be, so to say, extended to an infinite distance. § 2. Dkteemination of all Finite Groups of Linear Substitutions of a Variable. The problem of determining all possible finite groups of linear substitutions of a variable has been dealt with in various ways. * Cf. the numerous communications of Poincari in the " Comptes Rendus de I'Academie des Sciences," as well as his memoirs in Bd. xix. of the Math. Annalen, and in Bd. i. and ii. of Acta Mathematica (1881-83). Moreover, my essay in Bd. xxi. of the Math. Ann. (1882) may also be consulted: "Neue Eeitriige zur Riemann'schen Functionentheorie : " there, particularly, the litera- ture of the subject is noted and described in detail. + The latest researches of M. Picard move in this direction ; cf. Comptes Rendus, 1882-83, also Acta Mathematica, Bd. i. ii. SURVEY OF THE SUBJECT. 127 With my primary geometrical method* is connected the analy- tical method of fferr Gordan,'f then the general treatment by 31. C. Jordan, I by means of which he is in the position to solve the corresponding question for the case of a larger nnmber of variables. I shall here use a method of consideration, based on the function-theory, which I have already incidentally pointed ont.§ This starts from the idea of taking into consideration at once the equations, whose roots will be transformed into one another by the substitutions of the group, where it may easily be shown that these equations practically return to the funda- mental equations hitherto investigated. The process of thought, which M. Halphen has lately grappled with|| for a similar pur- pose, is not essentially different from the one here given. Moreover, a determination of all finite groups of linear substi- tutions of a variable is also implicitly contained in the investi- gations of Herr Fucks on algebraically integrable difierential equations of the second order, II investigations which we have already more than once cited in chapters ii. and iii., and to which we shall again pay regard in the following paragraph. We may say that these works of Herr Fuchs differ from mine in the fact that he brings forward the standpoint of the theory of forms quite at the beginning, while I commence with function- theory considerations. Let be the N linear functions, which, equated to x', represent a finite gronp of N linear substitutions of the variable x. Further, let a, h, be any two quantities, so chosen that none of the expres- sions i/r (a) are equal to h, or, what is the same thing, none * " Sitzvingsberichte der Erlanger physikalisch-medicinischen Gesellschaft of July 1874," Math. Annalen, Bd. ix. (1875). + "Ueber endliche Gnippen linearer Substitutionen einer Verauderlichen," Math. Annalen, Bd. xii. (1877). t "Mem. sur les liquations diff. lin. h int^grale alg^brique," Borohardt's Journal, Bd. 84 (1878); also "Sur la d^term. des groupes d'ordre fini contenus dans le groupe lindaire," Atti della Reale Accad. di Napoli (1880). § Math. Annalen, Bd. xiv. p. 149-150 (1878). II Loc. cit., p. 114. 11 Gottinger Nachrichten of August 1875 ; Borchardt's Journal, Bd. 81, 85 (1875-77). 128 GENERAL THEOREMS AND of tte expressions -^^ (J) are equal to a. We then form the equation : /I >, (■4'o (-r ) - g) ( -4/1 (g ) - g) (4-A,_i (x) - a ) _ ^ '^ (^'o W - b) (^1 (^) - 6) . . . - (v^^_, (x) - 6) - ^- Then we have evidently an equation of the iV^"' degree, which remains unaltered for the i\^ substitutions of our group, and whose N roots, corresponding to an arbitrary value of Jl, there- fore, in every case, proceed from one of themselves by our iV substitutions. In fact, if we substitute in (1), instead of x, any i/r (a;), the consequence is simply, since the yfr's by hypothesis form a group, that the factors in the numerator, and likewise the factors in the denominator of the left side of (1), are per- muted with one another in a certain manner. Our assertion shall now be this : that we shall be able to transform the equation (1) into one of the fundamental equa- tions hitherto considered by us by simply substituting for x and X appropriate linear functions of a; and X: _rix + f3 aX + h ■yx+ d' cX + d' To prove this, we first ask for what values of X the equation (1) may possess multiple roots. It is certain that if, for one value of X, one set of v a;-roots become identical, then all the corre- sponding a;-roots coincide in sets of v. This follows from the consideration of the substitutions i/r, just as we have proved the same theorem in the first chapter with respect to the groups of rotations, and those points on the sphere which remain fixed for certain rotations. We will now assume that to the values X=X-^, X^ . . . only i/j-tuple, I'j-tuple . . . roots correspond in the sense explained. According to the explanations of § 4 of our third chapter, we have then for the functional deter- minant of the (2iV— 2)"' degree, which is computed from the numerator and denominator of the left side of (1) [after we have turned both into integral functions of x by multiplication , N by the denominators of the i/^'s], — roots of multiplicity (i/j — 1), "i N — roots of multiplicity (v^—X), &c. Hence : SURVEY OF THE SUBJECT. 129 or otherwise written Our method will now first consist in considering this equation as a diophaniine equation for the integral numbers Vi, N, and seeking all the si/stems of solution thereof This latter is done in an extremely simple manner. We first agree that the number of the 1//3 cannot be less than 2, nor greater than 3 (inasmuch as we take iV, as is self-evident, to be > 1). Namely, if the number of the v^'s were equal to 1, the left side of (3) would be < 1, while the right side for iV> 1 is greater than or equal to 1. But if the number of the I'/s were ■>4, the left side of (2) would be >'2, because each element of the sum is itself ">„, and this would be no less a contradiction. We now first take the number of the i'/s equal to 2, and therefore simply write instead of (2) 0-.') ('H)*('-i)=(^4) or 1 1 2 Now it is self-evident that none of the v/s can be >iV; there- fore — > — We hence conclude that in the above case — and — must both be equal to -.r^- Hence we have (3) ^ = '2 = ^> where N is arlitrary ; and this is our first system of solution. Let us further take the number of the i'/s equal to 3, and therefore put, instead of (2), the equation Then I say, in the first place : at least one of the v^s must be equal to 2. Namely, if each of the three were j/j^S, the left-hand side I I30 GENERAL THEOREMS AND of (4) would be ^1, which is impossible. We therefore put, say, i/j = 2. For the remainder : 1 1-1 1 It is now possible that a second v, say V2, is equal to 2. We then find : 1-1 K,~N' Thus we have our second system of solution, which we will denote as follows, understanding by n an arbitrary number : (5) N=2u, .1 = 2, >,- . o But if neither of the two numbers v^, v^, is equal to 2, at least one of them must be equal to 3. For otherwise — y — would be '2 h ^-^, whereas it is to be >-^. Accordingly, let us put i'2 = 3. There remain : 1-1 1 Therefore anyhow 1-3 < 6. On the other hand, we can choose i'3 = 3, 4, 5, according as we wish. We get correspondingly N=12, 24, 60, and then in each case our conditions are all satisfied. There are therefore three more systems of solution, which we gather together into the following table : ( N=12, v, = 2, .2 = 3, ,3 = 3; (6) / N=2i, ., = 2, ►2 = 3, .3=4; ( N=60, Kj = 2, ., = 3, .3 = 5. The five different systems of solution so found correspond exactly, as we see at once, to our five fundamental equations : the bino- mial equation, the equations of the dihedron, tetrahedron, octa- hedron, and ikosahedron. We will now show that, according to the system of solution (3), or (5), or (6), which we like to attri- bute to our diaphantine equation, we can in fact in each case transforon our equation (1), in the way proposed, into the corre- sponding fundamental equation. SURVEY OF THE SUBJECT. 131 Let us take the case (3) to start with.. Instead of X we may introduce in it : We have then for Z^ and for Z= 00 an iV-fold root x. Our equation (1) therefore admits of being written as follows : and here we have only to put : in order to have before us the binomial equation : z'' = Z. In the other cases we will choose : X- X^ X^- X.2 so that for Z= merely Vj-tiiple, for Z=^ 00 merely j/3-tnple, for Z=\ merely i/j-tuple, i.e., double roots enter. Denoting by $1, $21 ^v appropriate integral functions of x, our equation (1) takes then the following form : Z:Z-\:\= 4./= (x) : 1. The functions of "l '2 *3 Herr Schiuarz arise on inserting in (18) for i/j, i/j, 1/3, any other three integers (whereupon - + - + - will be ^ 1). "1 "2 '3 In order to give a representation of the march of these func- tions, let us remark the following. In the chapter we have seen that, in virtue of our fundamental equations, the half-plane Z will be represented on spherical triangles of the 2-sphere, the angles of which are respectively -■> —> -• Just the same takes "l *2 "3 place in the case of the functions we are now speaking of, as soon as we have fixed upon the particular solution of (18), which we wish to take into consideration, and then develop this analytically. But while, corresponding to the algebraical character of the fundamental equations, a finite number of spherical triangles then sufficed to envelope the 3-sphere, now an infinite number of such triangles (no one of which infringes on another) are placed side by side. We must here distinguish when - + - + -is =lor <1. In the first case, all the spherical ''1 '2 '3 sides which bound the triangles pass, when produced, through a fixed point on the a-sphere, and we approach nearer and nearer to this fixed point as we multiply the triangles in succession, without, however, actually reaching it. The function Z has a finite value everywhere, except at this point. In the other cases, the bounding spherical lines have a common orthogonal circle, and this circle forms the limit which we approach, by increasing the spherical triangles, as near as we like, without, however, crossing it. Hence the function Z(z) exists only on one side of the orthogonal circle ; the orthogonal circle is for us what is described as a natural boundary.* As regards the corresponding group of linear substitutions, let us consider the spherical triangles in question alternately shaded and non-shaded. The group then consists of all linear sub- stitutions of 2 which change a shaded triangle into another shaded triangle (or a non-shaded into a non-shaded triangle). * Cf. throughout the memoir of Schwarz above cited, in which, moreover, appropriate figures are given. SURVEY OF THE SUBJECT. 143 Amongst the function thus introduced, the elliptic modular functions now form (to limit ourselves to the simplest kind) a special case, the case v-^ = '2, v^ = ?>, 1/3=00. The spherical triangle of the 2-sphere has then, corresponding to the value of 1/3, an angle equal to zero. If we allow the limiting circle, which Z{z) possesses on the s-sphere, to coincide with the meri- dian of real numbers, we are able to ensure that the totality of the corresponding linear substitutions is given by those integral, real substitutions , az + fi z = 5 yz + 6 whose determinant (aS — fiy) is =1. Let gq.,g^, be the invariants of a binary biquadratic form F(x.^., x^ (see § 11 of the second chapter), then it is known that A =5'2* — 27^3- is the correspond- ing discriminant. Now put the Z in question equal to the a 3 absolute invariant ■^. Then the z{Z) is 7iothing else than the ■ 3 A ratio of two primitive periods of the elliptic integral f- sjFj^) therefore the -=- of the Jacobian notation* It is impossible to enter here more minutely into the various relations thus touched upon. We will only bring forward this remark, that, in virtue of the conception developed, the elliptic modular functions appear, just in the same way as the expo- nential function and the cosine, as the last term of a series of infinitely many analogously constructed functions. Put in for- mula (18) fj throughout equal to 2, 1/3 equal to 3, and then let V2, beginning with 2, assume successively all integral posi- tive values. Then we have for 1/3 = 2 a case of the dihedron-f- (only that v^ is taken > v^, whereas we have usually elsewhere * See Dedekind in Borchardt's Journal, Bd. 83 (1877), also my essay "Ueber die Transformationen der elliptischen Functionen," &,c, in Bd. xiv. of the Math. Ann. (1878). t It is the same case to which, as explained in § 7 of the second chapter, the calculation of the double ratio of four points leads, or of the modulus of the elliptic functions, and which, moreover, will form a atartiug-point in the sequel for the solution of equations of the fifth degree. 144 GENERAL THEOREMS AND arranged the v'a in the order of the magnitudes), for v^ = 3, 4, 5, the tetrahedron, octahedron, and ikoaahedron, in order; then for greater values of v^ an infinite series of transcendant func- tions, whose termination for 7/3= oo is formed by the elliptic modular functions. § 7. Solution of the Tetrahedral, Octahedral, and Ikosa- HEDRAL Equations by Elliptic Modular Functions. Short as the preceding suggestions are, they suffice to make intelligible how it comes to pass that we can solve the equations of the tetrahedron, octahedron, and ikosahedron (or indeed the special case of the dihedral equation just mentioned) by means of elliptic modular functions. Let us first consider the logarith- mic solution of the binomial equation : or what is quite analogous, the trigonometric solution of the dihedral equation : 2" + ?-"= -4Z+2. Both equations admit of being regarded as a limiting case of a trivial algebraical solution, which consists in first calculating f from the equation : ^"'" = Z, or ^'"" + ^-'»»= - 4.Z- 2, understanding by m any positive integral number, and then finding 2 equal to a rational function of f : The transcendent solution proceeds from this on taking m = 00, whereupon f"*" is transformed in e^ just in the manner described, (_ ^n^^-mn jj^to 2 cos if, while z will become e". The case is precisely the same now tvith the representation of our fundamental irrationalities by means of elliptic functions. We convince ourselves, first, that each of the Schwarzian func- tions Vj, V2, v^, admits of being represented uniquely by means of every other i/j', v^, v^. In particular, therefore, if we limit ourselves to that series of functions for which i'i = 2, 1^2=3, the SURVEY OF THE SUBJECT. 145 only condition necessary for a single-valued representation will be that 1/3' is divisible by v^ But this is always the case if 1/3'= 00. All functions of our series, therefore, admit of a single- valued representation in terms of the elliptic modular ficnctions, and it is just this which is described as a solution of the equations in question by the help of the elliptic modular functions. I communicate here, without proof, the simplest formulae which present themselves in this direction for the tetrahedron, octahedron, and ikosahedron.* We write the three several fundamental equations, as we have always done, in the follow- ing manner : Y^"'^' T087«"'^' 1728/5 "■^- Then let ^, as just now, be the absolute invariant ^ of an elKptic integral of the first kind, -=^ the ratio of its periods, K' q = e . Then we have first, for the root of the octahedral equation, the simple formula : (19) ^=i^-^ — ; — CO this arises from the known equation : •^^'^{0, q)' on introducing (f, instead of q, on the right-hand side.f * Compare Bd. xiv. of the Math. Ann,, pp. 157, 158, also the essay of Signer Bianchi : " Ueber die Normalformen dritter und f unfter Stiife des ellip. Integ. erater Gatt.," and my own note : " Ueber gewisse Theilwerthe der 6-Functionen " in Bd. xvii., ibid. (1880-81). t This corresponds to the remark which we made above (p. 44 of the text) con- cerning certain researches of Abel's. In order to thoroughly understand the con- nection of what is to follow in this direction, let us compute for the biquadratic form (1 - a:-) (1 - Ic^n?) the absolute invariant -j-. We then obtain : (1 + 14^ + ^3)1 108F(l-i-2)<' and on inserting here for Vt the letter z, we have exactly the left side of the octahedral equation ; the symbols ^o, ^3, as also 3i, which I employ in the text, are the well-known Jacobiau ones. K 146 GENERAL THEOREMS AND We find, further, for the ikosahedral irrationality : (20) . = g---^ = 1^- ).^,^ ( , — CO an expression, therefore, which coincides with 9 when the term involving g^" is disregarded. The solution of the tetrahedral equation takes a rather more complicated form. We will in this case first replace the z hitherto used by a linear function of z, which vanishes at the summits of 'iP"=0, and becomes infinite at the opposite summits of $ = 0. In this sense we write : For the | thus defined we have then, first, the equation : (21a) z = g^'- = 64 (^'-^^' A "^3(^3 + 8)3' and, further, the transcendent solution : 2(-l)" (2k + 1) 33«'+3« (21b) f = - 6?^ . ,1^ 2(-1)''(6k + 1)2^"'+' — OD We have thus for our three equations severally determined one root ; we obtain the remaining corresponding roots if we -> • "■ iK' substitute in 5' = e ^°^ ~k^ *'^® infinite number of values : iK' „ iK' ' where a, 13, 7, S, are real integers of determinant 1. Sere all such systems a, ^, 7, 8, as coincide for modulus v^, or can be brought SURVEY OF THE SUBJECT. 147 into coincidence hy means of a uniform change of sign, always give rise to the same root* § 8. Formulae for the Direct Solution of the Simplest Eesolvent of the Sixth Degree for the Ikosahedkon. In accordance with the particular significance which we attach to the ikosahedral equation, the second of the formulae (19) — (21) of course has most interest for us. We have already explained that the simplest resolvent of the sixth degree which the ikosahedral equation possesses has been placed by Herr Kronecher in direct relation with the modular equation of the sixth order for a transformation of the fifth order of the elliptic functions (see § 15 of the preceding chapter). The formula in question has since been considerably simplified by Herr Kiepert and myself by the introduction of the rational invariants.-j- Since, in our researches on equations of the fifth degree, much regard will be paid to this very formula, it may also be com- municated here, the proof being left out, and the symbols used elsewhere being adapted. We have in § 15 of the preceding chapter (formula (46)) given the following form to the resolvent alluded to : ^6 - lOZ. ?3 + 12Z2 . ^ + 5Z2 = 0. Now let ^2) •^ b® ^^ invariants, already so denoted, of an elliptic integral, and let Z be taken = ■^^-. Further, let A' be that value which is derived from A by any transformation of the fifth order. Then the root of our resolvent is simply : (22) ?=-fJ. * V3 18 =3 for the tetrahedron, = 4 for the octaheuron, = 5 for the ikosahedron. For the special dihedral equation appertaining hereto, the same theorem would hold good for i-s = 2.J Compare for this, Mathematische Annalen, Bd. xiv. pp. 153, 156. t Cf. Bd. xiv. of the Math. Ann., p. 147, also Bd. xv. p. 86 (1878), and further, KiepcTt: "Auflosung der Gleichungen 5. Grades," and "Zur Transformations- theorie der elliptischen Functionen" (Borchardt's Journal, Bd. 57, 1878-79), and, finally, the memoir of Hurwitz just mentioned. 148 GENERAL THEOREMS AND If we like to express everything here in terms of K, iK\ and q respectively, and thereby to derive at the same time from each other the six different roots (22), we have first to insert the following values for g^ and ^-J^ -, (23)- oo and then for '^J' to put the following six values respectively : 00 oo .,. (24) f4icv ^ q 5 where i' = 0, 1, 2, 3, 4, and e is to be taken = e * . The indices 00, V, are here chosen exactly as in | 15 of the previous chapter. The formula (23) can at the same time be used to complete the data of the preceding paragraph ; namely, the absolute invariant of the elliptic integral is given by them in the form : 3 03 V 1 1 - r/' ^ ' A 1728o2 -^ JJa-Q'T § 9. Significance of the Transcendental Solutions. The significance of the transcendental solution with which we have now become acquainted is primarily a purely practical one. Logarithms, trigonometric functions, and elliptic modular functions have been long tabulated, in consideration of the importance which they possess in other fields of analysis. By reducing the solution of our equations to the said transcendent functions, we make these tables available for use, and avoid the tedious calculation which would be necessary in carrying out SURVEY OF THE SUByECT. 149 the method of solution by means of hypergeometrical series given in Chapter III* But there is a deeper conception of the transcendental solu- tions, by which the latter lose the foreign aspect which they seem to have in the midst of our other investigations, and rather become intimately combined with them. Let us consider, in order to fix our ideas, say, the solution of the ikosahedral equation as it is furnished by (20). As often as we subject to one of the infinitely many corre- sponding linear integral substitutions, the z, in virtue of this formula, experiences one of the 60 linear ikosahedral substitu- iK' tions. The group of the substitutions of - therefore appears isomorphously related to the groiip of the 60 ikosahedral suhstitu- tions. The isomorphism is only, if we may so express it, one of " infinitely high " merihedry : to the individual substitution of -j_- corresponds one, and only one, substitution of 2, while to every substitution of z correspond infinitely many substitutions of -„ . Let us now recall the considerations of § 5. Limiting ourselves there to finite groups of linear substitutions, we re- quired to bring into connection with one another such equation- systems generally (or form-problems) as are related to isomor- phous groups. We now extend this problem to infinite groups of linear substitutions, and recognise that our transcendent solutions realise special cases of the problem so generalised. We have obtained these solutions by making use of the theories, developed in other quarters, of certain transcendent functions. This is evidently a process which, in connection with our pre- sent considerations, is not theoretically satisfactory. We rather require a general treatment by means of which the develop- ments given in § 5, as well as our present transcendent solu- tions, will be furnished. Our reflections thus lead to a compre- hensive problem, which will embrace the theory of equations of * The unfortunate circumstance here arises, as regards the elliptic modular functions, that Legendre's tables for the calculation of the elliptic integrals have not been formed in a way which would correspond with Weieistrass' s theory of elliptic function. ISO GENERAL THEOREMS, ETC. a higher degree, as well as the law of constrnction of the 3- function. In proposing this problem, however, we have reached the limit, as in § 5, which bounds our present exposition, and which we may not pass.* * I will, however, not omit to call attention here to certain developments of M. Poincare's (on the general function which he denotes by Z) which behave just in the way here alluded to ; see Mathematische Annalen, Bd. 19, pp. 562, 563 (18S1). I have, further, to append here the following quotations, which resemble one another in relating to works in which, with greater or less completeness, the theories expounded in our iirst Part are connectedly dealt with: (1.) Puchta, "Das Oktaeder und die Gleichung vierten Grades," Denkschriften der Wiener Akademie, math.-phys. Kl., Ed. 91 (1879). This work might also be consulted throughout the following Part when we are concerned with the solution of equa- tions of the fourth degree (by means of the octahedral equation). — (2.) Cayley, "On the Schwarzian Derivative and the Polyhedral Functions," Transactions of the Cambridge Philosophical Society, vol. xiiL (1880). By the " Schwarzian deriva- tive " is there understood the differential expression of the third order, which we established in § 6 of the third chapter. — (3.) Wassilieff, "Ueber die rationalen Functionen, welche den doppeltperiodischen analog sind," Kasan (1880) (Russian). Herr Wassilieff there makes the interesting remark that Hamilton had already considered the group of the ikosahedral rotations with reference to their genera- tion from two operations. ("Memorandum respecting a New System of Non- Commutative Eoots of Unity," Philosophical Magazine, 1866). PAET II. THEORY OF EQUATIONS OF THE FIFTH DEGREE. CHAPTER I. THE HISTORICAL DEVELOPMENT OF THE THEORY OF EQUATIONS OF THE FIFTH DEGREE. § 1. Definition of our First Problem. The considerations of the preceding Part have given ns a deter- minate problem with regard to equations of the fifth degree. Now it would not be difficult to put the results, which I have to develop in this connection, as such in the foreground, and derive them in a deductive form. I intend, however, to avail myself of the inductive method here also, and this in such a way that, on the one hand, I pay regard to the historical development of equations of the fifth degree, and, on the other hand, make free use of geometrical constructions. I hope in this way to unfold to the reader not only the accuracy of definite results, but also the process of thought which led to them. In accordance with what has been said, our first task must, in any case, be that of giving an account and review of works hitherto published which are concerned with the solution of equations of the fifth degree, so far as these works will be used in the sequel. I shall here, for the sake of brevity, leave on one side all such developments as we shall not be immediately concerned with, however weighty and essential these may appear from more general points of view. To these belong, above all, the proofs of Euffini and Abel, by which it is established that a solution of the general equation of the fifth degree by extract- ing a finite number of roots is impossible ; and the parallel works, likewise set on foot by Abel, in which all special equa- tions of the fifth degree are determined, which differ in this respect from the general equation. To these again belong the efforts of Hermite and Brioschi to apply the invariant theory of binary forms of the fifth order to the solution of equations of the fifth degree ; not that the use of the invariant-theory pro- 154 THE HISTORICAL DEVELOPMENT OF cesses will be altogether dispensed with in the following pages, only that in our case these relate throughout, as in the preced- ing Part, to such forms as are transformed into themselves by determinate linear substitutions, and not to binary forms of the fifth order. Finally, we leave on one side the question of the reality of the roots of equations of the fifth degree ; in particular, therefore, the extended investigations by which Sylvester and Hermite have made the reality of the roots to depend on the invariants of the binary form of the fifth order. If we limit our task in the manner here described, there remain two fields of labour which we have to consider. The object of both of them is to study the roots of the general equa- tion of the fifth degree as functions of the coefiScients of the equation. Both start by simplifying the functions in question, so that, instead of the five independent coefficients of the equa- tion, a smaller number of independent magnitudes will be intro- duced. Now the means which are employed for this purpose are diflferent ; in the one case it is the transforviation of the equations, in the other it is the construction of resolvents. The method of transformation goes back, as we know, to Tschirnhans.* Let (1) x" + Ax''-'^ + Bx''-^+ . . Mx + N=Q be the proposed equation of the w"' degree; then Tschirnhaus put: (2) y = a. + j3x + yx^ + . . /u. . a;" " i, whereupon, by elimination of x between (1) and (2), he ob- tained an equation for y, also of the Ji*"" degree, to which he endeavoured to impart special properties by a proper choice of the coeflBcients a, /3, y, . . . We will at once describe the results which have been found by this assumption for the special case of the equation of the fifth degree. Let us first agree that with the y'a the x's are also found, at least so long as the equation for the y's, as we manifestly assume for the equation (1), possesses different roots. For in this case the equations (1) and (2) [in which we now consider y as the unknown magni- * " Nova methodus auferendi omnes terminos intermedios ex data aequatione," Acta evuditorum, t. ii. p. 204, &c. (Leipzig, 1683). The title itself shows that Tsuliirnbaus realised (as Jeriard did later ou) the range of his method. EQUATIONS OF THE FIFTH DEGREE. 155 tude] have only one root x common, and this x can therefore be rationally calculated by known methods. The method of the construction of resolvents has also long ago been employed for the solution of equations of the fifth degree. Notable in this respect is the year 1771, in which Lagrange, Malfatti, and Vandermonde, independently of one another, published their closely related investigations.* How- ever, the results which these attained rather served to point out the existing difficulties than to remove them. Herr Kro- necker first succeeded, in 1858, in establishing a resolvent of the sixth degree for the equation of the fifth degree, by which a real simplification was efiected.-|- We shall have to limit ourselves in our further account, so far as the construction of resolvents is concerned, to the exposition of Kronecker's method and the further researches connected with it. The two fields of labour which we have thus placed beside one another are concerned, per se, with purely algebraical pro- blems. Howbeit the development of analysis has entailed their both appearing intimately connected with the more extended problem: to effect the solution of equations by the help of proper transcendent functions. We have shown in the last chapter of the preceding Part:J: that such a use of transcendent functions is primarily of merely practical value, and should not be confounded with the theoretical researches on the theory of equations. However, we must not neglect in the following account to consider the different methods by which the solution of equations of the fifth degree has been specially connected with the theory of elliptic functions. For it has been just these methods, as we have already suggested, which have led to a tighter grasp of the purely algebraical problems also. * Lagrange: " Rdflexions siir la resolution alg^brique des ^nations,'' Mu- ufioires de I'Acaddmie de Berlin for 1770-71, or CEuvres, t. iii. Malfatti: "De cequationibus quadrato-cubicis disquisitio analytica," Atti dell' Accad. dei Fisiocritici di Siena, 1771 ; also, "Tentativo per la risoluzione delle equazioni di quinto grado, ihid., 1772. Vandermonde: "M^moire sur la resolution des equations," Memoires de I'Academie de Paris, 1771. t Compare the later references. X I shall in future denote references to the preceding Part by letting the number of the chapter, represented by an arable number, succeed the romau number I. (/. therefore in this case I. 5, § 7, 9. 156 THE HISTORICAL DEVELOPMENT OF For the rest, let it be observed that there is no essential anti- thesis between the two fields of labour which we are contrast- ing. If we have succeeded in turning a proposed equation of the w"* degree by transformation into another which contains a smaller number of parameters, we can afterwards derive re- solvents from the latter, and consider these as peculiarly simple resolvents of the original equation ; or conversely, if we have come into possession of a special resolvent of the initial equation by any of these methods, we can return from it, by renewing our formation of resolvents, to an equation of the w"" degree, which latter will then admit of being transformed directly from the proposed equation. § 2. Elementary Eemaeks on the Tschiunhausian Transformation — Bring's Form. In order to compute the equation of the n"" degree which the y's of formula (2) satisfy, it is most convenient to compose its coeflBcients directly, as symmetric functions of the y's, from the symmetric functions of the x'b. In this way we recognise at once that the coefficient of 2/""* is an integral homogeneous function of the /e"" degree of the indeterminate Tnagnitudes a, /8, 7, . . . V. Hence we have a linear equation with n unknowns to solve, if we wish to expel the term involving y"^^ from the transformed equation, and a quadratic equation of the same kind appears in addition if the term involving y"~^ is to vanish also. We satisfy both these equations together if we consider «— 2 of the unknowns as parameters, and determine one of the remaining ones by means of a quadratic equation after eliminating the last unknown. I shall describe an equa- tion in which the terms involving «/"""*, y"~^., are wanting as a principal equation for the future. The Tschirnhausian trans- formation, therefore, allows us to reduce every equation to a principal equaiion with the help of merely a square root. On the other hand, we meet with difficulties as soon as we require that another term in the equation of the y's shall vanish. In fact, we then come upon elimination-equations of a higher degree, which we do not know how to treat by elementary means. It is here that a more searching investigation has EQUATIONS OF THE FIFTH DEGREE. 157 brought to light an important and — for our future exposition — fundamental result. The equation of elimination of which we speak will be of the sixth degree if we require that the terms ^n-i^ 2/"~^, 2/"~') shall vanish simultaneously ; it has been shown that ly proper choice of coefficients of transformation for w>4 the said equation of the sixth degree can he reduced to an equation of the third degree hy the solution of a quadratic equation. The result thus described is usually ascribed to the English mathematician Jerrard, who made it known in the second part of his Mathematical Researches (Bristol and London, 1834, Longman). But it is of much earlier date so far as equations of the fifth degree are concerned. As Hill remarked in the Transactions of the Swedish Academy, 1861, it had already been published in 1786 by E. S. Bring in a Promotionschrift sub- mitted to the University of Lund.* I should, nevertheless, have retained in the following pages the practice, generally diffused at the present time, of describing it in connection with Jerrard, had not the latter in his works relating to this matter brought forward, amongst some interesting results, a lot of thoroughly false speculations : he believed (just as Tschirnhaus did) that he could remove, by the help of his method, all the intermediate terms, not only from equations of the fifth degree, but equations of any degree, by means of elementary processes, and did not lay aside this view in spite of incisive refutation from the other side.t I shall therefore in future speak of * The full title runs : " Meletemata qusedam mathematica circa transforma- tionem aequationem algebraicarum, quae preside E. S. Bring . . . modeste sub- jicit S. G. Sommelius." We might, perhaps, have been led by the title to suppose that Sommelius was the author, but I learn from Herr Backlund of Lund that would certainly be erroneous, inasmuch as the Promotionschriften were then composed entirely by the candidates, and only served the examiners as a substratum for disputation. The principal points of Bring's treatise are reprinted in the communication, already mentioned, of Hill to the Swedish Academy, and again in the Quarterly Journal of Mathematics, vol. vi., 1863 (Harley, "A Contribution to the History," &c.), and, finally, in Grunert's Archiv, t. xli., 1864, pp. 105-112 (with remarks by the editor). + Jerrard's further publications are found principally in the Philosophical Magazine, vol. vii. (1835), vol. xxvi. (1845), vol. xxviii. (1846), vol. iii. (new series) (1852), vols, xxiii., xxiv., xxvi. (1862-63), &c., and are, therefore, for the most part, later than the report (as lucid as it is voluminous) which Hamilton fur- nished in 1836 for the British Association for the Advancement of Science (Reports of British Association, vol. vi. , Bristol). Further, CocUe and Cayley repeatedly opposed the assertions of Jerrard (Phil. Mag., vols, xvii.-xxiv, 1859-62). iS8 THE HISTORICAL DEVELOPMENT OF Bring's equation. Let us write the principal equation of the fifth degree (as it is to be written henceforward) in the follow- ing form : (3) 1/ + 5a?/2 + 5&2/ + c = 0. Then it will be to the purpose to retain the coefficient 5 in Bring's form also. On substituting at the same time z for y, for the sake of distinction, we have : (4) 25+5fe + c = 0. Bring's equation still contains, as we see, at first two coeffi- cients. "We can, however, at once remove one of them by- putting z = pt and then suitably determining p. We can, there- fore, by a proper Tschimhausian transformation, efiect that the five roots of the equation of the fifth degree shall appear to depend on a single variable magnitude. This result is more peculiarly important because we are much more completely masters of the functions of a single argument than of those of a larger number of variables. Let us write (4), e.g., as follows (as Hermite has done in his researches to be quoted immediately) : (5) fi-t-A = 0, then it is very easy, on the one hand, to exhibit by Riemann's method how the five roots t depend on A, and, on the other hand, to establish for any values of A appropriate developments in ascending and descending powers which allow the five roots t to be computed to any approximation. When we have thus become acquainted with Bring's result, we may postpone a deeper penetration into its basis, and also a criticism on its significance, till later, when we shall have fur- ther occasion to do so in connection with our own developments. I also omit to enumerate all the numerous commentaries which the researches of Bring and Jerrard respectively have received in the course of years. One of the first expositions of this method, and, at the same time, the one most widely known, is perhaps that in Serret's "Traits d'algfebre superieure" (1st edition, 1849). Hermite has also dealt with Bring's transfor- mation,* aiming, however, as already observed, at the appli- * In the coEnprehensive treatise (which will be often mentioned) : " Sur I'dquation du einqui^me degrd," Comptes Rendns, t. Ixi., Ixii. (1865-66). Cf. particularly t. Ixi. pp. 877, 965, 1073, t. Ixii. p. 65. EQUATIONS OF THE FIFTH DEGREE. 159 cation of the invariants of binary forms of the fifbh degree; we must remark that Hermite has determined the irrationali- ties necessary for the transformation mnch more completely than is usually done. § 3. Data concerning Elliptic 'Functions. The special questions in the theory of elliptic functions on which we must now inform ourselves lie in the region of the theory of transformation. With the usual notation let k be the modulus of an elliptic integral : /dx \ the modulus which results from a transformation of the w"" order, where n is to denote an uneven prime number. Then, according to JacoM* and Sohnkef respectively, there exists between s/ k = u and v'\ = v an equation of the (w+l)"' degree in each of these quantities, the so-called modular equation: (6) /(m, «) = 0, which, e.g., for 71 = 5, runs as follows : (7) m6 _ i;6 + 5mV (m2 - ^2) + 4My ( 1 - M V) = 0. Here u may be expressed ia various ways in terms of 17 = 6 "a', e.g., as follows : (8) 1 "N'„2m« + m we obtain the («+l) values of v, which satisfy the modular equation on inserting in this formula in order : 1 1 n-l 1 2tir where a = e " - The modular equation therefore gives us an example of an equation with one parameter which can he solved * "Fundamenta nova theorise functionum ellipticarum" (1829). + " jEquationes modulares pro transformatione functionum ellipticarum " (Crelle's Journal, t. xii., 1834). i6o THE HISTORICAL DEVELOPMENT OF hy elliptic modular functions* The parameter is v, ; we find from it the corresponding q on reversing the formula (8), or calculating the magnitudes K and K' from (5) : where k'^ = 1 — i^. The (n + l) roots v are then obtained by means of the substitutions (9). We now ask whether it is not practicable to effect, by the help of the modular equation, the solution of other equations also. To this end we shall have, above all — in accordance with the explanations which we have given in § 1, 4 — to determine the group of the modular equation. This is what Galois himself has already accomplished.f Corresponding to the substitutions (9), Galois denotes the roots of the modular equation by the following indices : (11) Voo, V„ ■«!, . V„_i. If we then disregard mere numerical irrationalities,! the group of the modular equation is formed of those permutations of the vJb which are contained in the following formula : (12) .' = ^^niod.(»), which we have already considered above in special cases (I. 4, § 15 ; I. 5, § 7). The coeflBcients a, yS, 7, S, are here otherwise arbitrary integers which satisfy the condition (a8 — ^y) ^ 1 (mod. n). We interpret this result specially for n=5. The group (12) will then be, as we saw before, holohedrically isomorphous with the group of the 60 ikosahedral rotations, i.e., expressed in the abstract, with the group of even permutations of five things. We hence conclude that the modular equation (7) possesses resolvents * We have already become acquainted with other examples above, I. 5, § 7, 8 ; Bince, however, we have here to explain the historical development of the theory, these are for the present not considered. t See "CEuvres de Galois," Liouville's Journal, t. xi. (1846). + According to the researches of Hermite, the single numerical irrationality / Ezl here coming under consideration is V (-1) 2 . n. Cf. the exposition in C. Jordan's " Traits des substitutions et des equations alg^btiqaes," p. 344, &c. EQUATIONS OF THE FIFTH DEGREE. i6i of the fifth degree, whose discriminant, after adjunction of a numerical irrationality ( V5, according to Hermite), is the square of a rational magnitude. Will it be possible to put the general equation of the fifth degree, after adjunction of the square root of its discriminant, in connection with such a resolvent by means of a Tschirnhausian transformation ? Or, conversely, shall we be able, after adjunction of the square root of its dis- criminant, to establish a resolvent of the sixth degree of the general equation which proceeds from the modular equation (7) by appropriate transformation? These are just the two ways of attacking the solution of equations of the fifth degree by elliptic functions which have been taken in hand and worked out by Hermite and Kronecker respectively. Before we enter on an account of their results, we have an important addition to make from the theory of elliptic functions. We mentioned just now the idea of subjecting the modular equation itself to a Tschirnhausian transformation. This has already been done in a certain form by Jacobi, who placed alongside of the modular equation (6), properly so called, a series of other equations of the (71+ 1)'** degree which can replace it. It is no part of my plan to communicate a rational and comprehensive theory of the infinite number of equations which thus come under consideration.* We must confine our thoughts to an especially important result which Jacobi had estabUshed as early as 1829 in his " Notices sur les fonctions elliptiques." f Jacobi there considers, instead of the modular equation, the so-called multiplier-equation, together with other equations equivalent to it, and finds that their (n+1) roots an n + \ composed in a simple manner of —^ elements, with the help of merely numerical irrationalities. Namely, if we denote these elements by \, \, . . . A„-i, and, further, for the roots z of the equation under consideration, apply the indices employed by Galois, we have, with appropriate determination of the square root occurring on the left-hand side : * Cf. for this, so far as modular equations proper are concerned, my develop- ments : "Zur Theorie der elliptischen Modulfunctionen," in Bd. xvii. of Muthe- matische Aunalen (1879). + Crelle's Journal, Bd. ui. p. 309, or Werke, t. i. p. 261. L i62 THE HISTORICAL DEVELOPMENT OF v/2»=^(-l)^.9J. Ao, s/2„ = \ + £i-Ai + C*''A2 + . . . i- '^ ' " ■ A„_i (13) 2iir for i'=0, 1, . . . («— 1) and e = c'"- , so that, therefore, the fol- lowing relations hold good between the V«'s : (14) where iV is to denote any one of the — h— non-residues for modulus n. Jacobi has himself emphasised the special significance of his result by adding to his short communication : " C'est un theoreme des plus importants dans la th^orie algebrique de la transforma- tion et de la division des fonctions elliptiques." Our further report will show how true this remark has proved. In the hands of Kronecker and Brioschi, the formulse (13) (14) have attained a general importance for algebra, inasmuch as the savants just mentioned determined to consider Jacobi's equa- tions of the (w.-l-l)"' degree, i.e., therefore equations whose {n + l) roots satisfy the established relations, independently of their connection with the theory of elliptic functions.* But in particular, on the existence of the Jacobian equations of the sixth degree (which correspond to n = 5), rests Kronecker's theory of equations of the fifth degree, as we shall soon have to show in detail. § 4. On Hermite's Work of 1858. We have now all the preliminary conditions for understand- ing Hermite's first work in this connection, the oft-mentioned * I follow throughout the notation and nomenclature of Signer Brioschi, as I did in my earlier publications. Herr Kronecker differs particularly in writing s=/-, and thus obtaining equations of the (2»i + 2)"' degree, whereupon linear identities corresponding to the formulae (14) exist between the magnitudes/. I do not see that this possesses many advantages. EQUATIONS OF THE FIFTH DEGREE. 163 memoir of April 1858.* Hermite had even earlier been con- cerned (as also had Betti) with the proof of Galois' data con- cerning the group of the modular equation. But the object was, so far as the case « = 5 was concerned, to actually esta- blish, in the simplest form, that resolvent of the fifth degree which the modular equation (7) ought to possess. This is what Hermite now attained to, when he put : (1 5) 2/ = («'. - »o) («i - "4) ("2 - ^'s). and found the following corresponding equation of the fifth degree z : (16) t/-2*.0K U* (1 - mS)2 . 2/ - 26 ^/5» . tfi (1 - 2(8)2 (1 + „S) = o.t We have here exactly the Bring form with which we became acquainted above, and, in fact, it is easy to identify any Bring equation with (16) by a suitable choice of u. It is sufficient to return to the simplified form which we communicated in (5) : We reduce (16) to this form on taking : (17) 2/= 2^53 . u . vr^s _ t^ the coefficient A will then be equal to the following expres- sion: 2 1+mS (18) ^p • m2(i_j,S)»- ' and here we determine u from A the more easily in that we have to do with a reciprocal equation with regard to u. Heuce the solution of any Bring equation is furnished by the formulas of Hermite, and with it indirectly the solution of the general equation of the fifth degree by means of elliptic functions. Hermite's work has, as follows from this short account, no kind of relation to the algebraical theory of equations of the fifth degree. Rather it moves throughout in the field of elliptic modular functions, and, moreover, the series of further researches which Hermite has published on the theory of modular func- * Coniptes Rendus, t. 46 : " Sur la resolution de I'dquation du cinquifcme degrd." t For the proof cf., say, Briot- Bouquet, "Theorie des fonctions elliptiques" (Paris, 1875), p. 654, &c. i64 THE HISTORICAL DEVELOPMENT OF tions took its origin in these. This is the reason why Hermite's solution of the equation of the fifth degree only comes cursorily under consideration in our following exposition ; for the use of elliptic functions appears altogether secondary to the con- ception which we shall henceforward maintain. This would, of course, be at once changed if we wanted to take into account in detail the general ideas which we formulated in the conclud- ing paragraph of the preceding Part, a course which must be deferred to future expositions. Together with Hermite's first work we advisedly mention two communications of Brioschi and Jouhcrt, who both compute the resolvents of the 5th degree for the multiplier-equation of the 6th degree (a special Jacobian equation, therefore, of the 6th degree), and hence likewise obtain the equation (16).* Kro- necker had also, as he informs Hermite, dealt with resolvent construction of this kind.-}- § 0. The Jacobian Equation of the Sixth Degree. Continuing our account, let us now first turn our thoughts to the researches which Brioschi and Kronecker have made on the Jacobian equations of the sixth degree.;}: Let us first remark the following facts. Whenever two investigators have worked at the same subject simultaneously and in relation to one an- other, it is difficult to distinguish what first issued from the one, what from the other. The chronological method, which refers to the dates of the individual publications, is certainly not always accurate ; but it is, after all, the only one which can be handled with any certainty. In this sense we shall now proceed on the basis of this method. 1 begin with recounting the works which Signor Brioschi has published in the first volume of the Annali di Matematica, Serie I. (1858). * Brioschi : " Sulla risoluzione delle eqnazioni di quinto grado " (Annali di Matematica, Ser. I. t. i., June 1S58), Joubert in a communication from Her- mite in voL 46 of the Comptes Rendus ("Sur la rdsolution de I'dquation du quatrieme degrd," April 1858). See also Joubert : " Note sur la r&olution de I'equation du cinquiome degrd," in the Comptes Rendus, t. 48 (1859). + Letter to Hermif-e, June 185S. See Comptes Rendus, t. 46. t Compare the exposition of this relation by Jlcnniie in his memoir, already mentioned: " Sur I'equatiou du cinquieme degr(5," Comptes Itendus, particularly t. U2 (1866), pp. -245-247. EQUATIONS OF THE FIFTH DEGREE. 165 After Signor Brioschi had first proved * (1. c.) the data of Jacobi, he concerned himself with the actual establishment of the general Jacobian equation of the sixth degree. His result is as follows.-f- Let A^, Aj, A2, be three magnitudes which occur in (13) corresponding to w = 5; further, let: B= 8A/A,A2 - 2Ao^Ai=A/ + A.^A/ _ A,(A,^ + A./), (19) \ C= 320A(,»Ai2A22 - leOA/AjSA..^ + 20A„2Ai^A24 + GAi^A/ - 4Ao(Ai5 + A,5) (32Ao'' - 20A„=AiA., + SAj'^A.;) + Ajio + Aji". ' Then the general Jacobian equation of the sixth degree will be the following : (20) (z - Af - 4:A(z - A)^ + lOB{z -Ay-C{z-A) + (SIP -AC) = 0. Brioschi further seeks to construct a resolvent of the fifth degree as simple as possible for this equation, and to this end first J puts (following Hermite's example) : (21) y = (z.^-z,)(z,-z^{z,-z.^, but then remarks, in connection with a letter of Hermite's, that the square root of this expression is already rational in the A's, and gives rise to an equation of the fifth degree.§ Let x be this square root ; then Brioschi finds for the five values of which X is susceptible the following formuliB : (22) x,= - Mi(4A„2 - A.A^) + -^iiA.lK^^ - A./) + 63''( - 2AoA/ + AjS) + f*''Ao(4Au2 - A^A,), while for the corresponding equation of the fifth degree he finds this: (23) 3fi+lOB3^ + 5{9B^-A(T)j:-y^ = 0, where 11 is the discriminant of the Jacobian equation (20). || The multiplier-equation of the sixth degree for elliptic func- tions (to which Jacobi'a remark first related) is of course con- tained in (20) as a special case. Brioschi finds that it is * P. 175, I. c. (May 1858). t P. 256, 1. ^. {June 1858). i Loo. cit. § P. 326, 1. c. (Sept. 1858). II I have here, in opposition to the original formula of Brioschi, given the numerical coefficients, as Joubert had done later on ("Sur I'equation du sixifeme degrc," Comptes Rendus, t. 64, 1867). i66 THE HISTORICAL DEVELOPMENT OF essentially characterised by the condition B = Q, whereupon (23) becomes a Bring equation. To Herr Kronecker is due the credit of first directing his attention to the case A = Q, and also of effecting its solution by means of elliptic functions. We need not communicate here in detail his primary formulse as he noted* them in his letter to Hermite, and as Brioschi then proved them in the memoir (to be presently described more in detail) in the first volume of the Atti of the Istituto Lombardo.-f For they are considerably simplified if, instead of the modulus k (which Herr Kronecker used), the rational invariants of the elliptic integral g^, g^, A, are introduced, and we have already become acquainted (I. 5, § 8) with the formulse of solution in question in this simplified form. In fact, the Jacobian equation of the sixth degree loith A = is none other than that simplest resolvent of the sixth degree lohich ive have established in I. 4, § 15, in the case of the ilcosahedron. We have only to put : and correspondingly : (25) B=-f,C=-H. At the same time, for A-=Q, the resolvent of the fifth degree (23) is transformed into the following : (26) a;5+io2j^ + 4552^_^ /n^O, which agrees with formula (27) of I. 4, § 11. I mention these relations only cursorily, to return to them later more in detail. It remains to consider one final direction of investigation with regard to the Jacobian equations of the sixth (or indeed of any) degree, that which Herr Kronecker first took in hand J in his algebraical communications from the year 1861 onwards, and which was then followed up further by Signer Brioschi in particular in the first volume of the second series of the Annali di Matematica (1867). § The object is to construct from one * Comptes Rendus, t. 46, June 1858. t " Sul inetodo di Kronecker per la risoluzione dalle equazioni di quinto grado " (Nov. 18.58). t Monatsberichte der Berliner Alcademie. g "Lasoluzione piu generale delle equazion! del 5. grado." See also "Sopra alcune nuove rdazioni modular!," in the Atti della R. Accademia di Napoli of 1866. EQUATIONS OF THE FIFTH DEGREE. 167 Jacobian equation a new one by a Tschirnhausian transforma- tion. Herr Kronecker remarks that this is possible in two ways, inasmuch as the roots Za,,Zy, of the transformed equation (which correspond to the z^, z^, of the original equation) either just satisfy the formulae (13), (14) (where e can be replaced by 6^ at pleasure, understanding by iJ a quadratic residue of n ; this only signifies a change in the order of the roots) ; or they satisfy those others which proceed from, (13), (14), on replacing e hy e^, where I^ is to denote an arbitrary non-residue to the modu- lus n. Let n be, as we will now assume, equal to 5 ; then we can in the first case put 'JZ equal, for example, to -— ^ or equal to — — ; the most general expression for 'JZ here coming under consideration arises on combining sjz and the two mag- nitudes mentioned multiplied by arbitrary constant factors : (27) Vz=x. ./i + ^ ^f + -T# OA 013 We solve the second case on first constructing for it a particular example, which is furnished, say, by : 1 C afterwards we treat the Jacobian equation corresponding to this example exactly according to formula (27). We shall return later on more in detail to the principle of these transformations. Meanwhile let us find room for the following remark. If we calculate for the 'JZ of formula (27) the expression A, this will be an integral homogeneous function of the second degree of the X, /i, V. We can make this zero by, for instance, putting i/ = and determining X : /i by means of the resulting quadratic equation. We can therefore hy mere extraction of a square root transform the general Jacobian equation of the sixth degree into one with A = 0. Signor Brioschi has since collected* his researches here in- dicated, as also the further ones to be described presently, which relate specially to the theory of equations of the fifth * "Ueber die Aufliisung tier Gleichuiigen fiinfteii Grades" (1S7S). i68 THE HISTORICAL DEVELOPMENT OF degree, in Bd. xiii. of tte Mathematisclie Annalen ; and they are all the more welcome because his original publications, widely scattered as they were, could have been only with diffi- culty accessible to many mathematicians. Herr Kronecker has also since returned to the theory of the general Jacobian equa- tions,* but the questions there treated by him lie beyond the limits which are prescribed for our present exposition. § 6. Kkonecker's Method for the Solution of Equations OF THE Fifth Degree. Having premised the theory of the Jacobian equations of the sixth degi'ee, we can with ease describe the nature of that method of solution which Herr Kronecker has developed in his oft-cited letters to Hermite (Comptes Rendus, t. 46, June 1858) for the general equation of the fifth degree. The Jacobian equations of the sixth degree are very intimately bound up with the theory of elliptic functions, but they also represent, as we have already remarked (and this in virtue of formulae (13), (14)), a remarkably simple type of algebraical irrationalities per se. Herr Kronecker's particular discovery is this : that from the general equation of the fifth degree after adjunction of the square root of the discriminant, rational resolvents of the sixth degree can he established lohich are Jacobian equations. To this is appended the further remark, which we led up to just now : that we can transform the Jacohian equation in question by the help of only one additional square root into one with ^ = 0, therefore into a normal form icith only one essential parameter,^ which admits of solution by elliptic fuiictions. In Herr Kronecker's original communication the two points here separated are, however, not clearly distinguished. Herr Kronecker limits himself to communicating the following rational function of the five roots of an equation of the fifth degree : * Monatsberichte der Berliner Akademie of 1879: "Zur Theorie der ajge- braisoht'n Gleichungen. '' + Here again we reduce it to only one parameter by putting z=pt and deter- mining p suitably. EQUATIONS OF THE FIFTH DEGREE. 169 J V! *0' ^V *2' ^3' •''4/ n=4 (29) _ >rn >r-T . 2mr. 22,3 \ ~ ^ ^ sin — ^^iK„,a; „^.„a; ^^2n ■'"''•* m^+ 1^111+ 2nj) \^ n=0 in which he supposes v so determined that 5'/^ = 0, and then remarks that the several fa, which arise from (29) by even permutations of the x's, satisfy an equation of the twelfth degree of the following form : (30) /i^-lO-^./e + S-v^^^-vl/./^, which will admit of solution with the help of elliptic functions. Here (30), provided we put/^ = 3, is the Jacobian equation with ^ = 0, and the vanishing of A corresponds to the vanishing of We are indebted to Signor Brioschi for having made the deeper meaning of Kronecker's method accessible to the mathe- matical public in a lucid and at the same time a more general form, and this in the memoir mentioned just now : " Sul metodo di Kronecker,'' &c., in the first volume of the Atti of the Istituto Lombardo (Nov. 1858). We do not here recur to the contributions which Brioschi has there made to the general theory of the Jacobian equations of the sixth degree. What here interests us is that he establishes a general rule of construc- tion/or the roots z, ofivhich a special case occurs in formula (29). Let: (31) v{x^,x^,X2,x.^,x^ be a rational function of the five x'& which remains unaltered for the cyclic permutation : 1^0' ■''l' •'-V ^S,^ ^i) further, let : (32) v' = v (x^, x^, a-j, x.„ a-,). Brioschi then puts (33) v-v=u^ and derives from this function five new functions u^, Mj, ii^, "3, '>t^, by first subjecting the x's to the substitution 170 THE HISTORICAL DEVELOPMENT OF and then bringing into application the cyclic permutation already mentioned. Tlien the following expressions are in general found to he the roots of a Jacohian equation of the sixth degree, which remains unaltered for all even permutations of the x's, and hence possesses as coefficients ratioTial functions of the coefficients of the equation of the fifth degree and of the square root of its discri- minant : (34) 2o = ^1 = u^ Jo + Mq + Mj + M, + ?(3 + u^)\ U — Mn + Mj \/5 - Mj + «3 ■u,)\ et^ + !«Q — 11^ + ^2 V 5 - M3 + M4)^, , + Mfl + Mj - Mj + "3 '^S - 1*4)^, ■ Mj + Mj + M2 - W3 + W4 n/5)2. These formulae become still more concise if we note the elements A(, ,Ai, Aj, of which the Vs's, in accordance with (13), are com- posed. The comparison gives simply : Ao n/5 = U^ -Jh + ? = .ejPi + SiPi, <1 = fiQi + f2'22. '■ = ?i^''i + ?2^^2' or (16) p = ?iPi + s,P.2 + faPa, &c. If we then introduce these values into the equation of the surface, or the equations of the curve, as the case may be, we obtain for p^ : p^ an equation, or for pj : pg • Ps ^ system of equa- tions of the «."' order ; each root of this equation or of this system of equations (as the case may be) gives us a Tschim- hausian transformation of the required properties. The irra- tionality which is thus required for the production of the transformation is evidently in general an accessory one. For there is no a priori reason why the discrimination of the 7i-points of intersection of an arbitrary straight line with the surface, or of a plane with a curve, should have anything to do with the distinction of the collineations which transform this surface or curve into themselves. It need hardly be said that the general process thus described, INTRODUCTION OF GEOMETRY 189 practically speaking, does not take us far. If we tried to treat the different special cases (enumerated in § 3, 4) of equations of the fifth degree by its means, we should be brought at once after the first two cases to auxiliary equations of higher degree than the fifth. We vnll therefore in the sequel only use our general process, or siippose it tuted, in order to transform the general equation of the fifth degree into a principal equation. In fact, we shall afterwards (in the fifth chapter) bring forward proof that in this special case the general process cannot be improved, inasmuch as it is in no way possible to get rid of the accessory square root which is introduced by our process. On the other hand, we shall succeed in all the other cases in find- ing more simple methods for the production of the transforma- tion. These methods were partly touched upon in the develop- ments of the preceding chapter ; we add here a few supple- mentary remarks. First, as regards Bring's transformation, we have stated already that it is possible, instead of the original system of equations of the sixth degree with which we have to deal, to substitute a sequence of quadratic equations and a cubic equa- tion. We can now, in reliance on our geometrical method of representation, express this much more precisely. The theory is marshalled in detail as follows. We first of all transform the general equation of the fifth degree, in the way above described, into a principal equation (where we require a first square root and an accessory square root). But then arises, geometricalbj, the important fact that through every point on the principal surface pass two linear generators thereof, of which each meets Bring's curve in only three other points. We shall therefore, in order to pass from an arbitrary point on the principal sur- face to a point on Bring's curve, first require another square root in order to define the generator passing through the point, and then, in fact, obtain an equation of the third de- cree, which determines the points of intersection of the chosen generator with Bring's curve. It has been already stated that we shall bring forward later on (in fact, in the third and next chapter) explicit formulte for all the steps required for Brink's theory. We only observe here, therefore, what was for the most part passed over in laying down the theory, that the 190 INTRODUCTION OF GEOMETRY. second square root (which defines the two generators of the principal surface) is not an accessory one, but coincides with the square root of the discriminant of the equation of the fifth degree. The irrationality which will be introduced by the cubic auxiliary equation is, on the contrary, again an accessory one ; the cubic equation is also in Galois's sense general, i.e., such as possesses a group of six permutations. We discuss, moreover, the equations of the fifth degree brought forward by Brioschi, which depend on the Jacobian equations of the sixth degree. By the existence of Kronecker's resolvent a method is indicated, as we remarked before, of transforming the general equations of the fifth degree into these special ones. In the first place, we have here to deal with the diagonal equa- tion of the fifth degree. Our previous account shows that only the square root of the discriminant, and therefore in no way an accessory square root, is required in order to turn the general equation of the fifth degree into a diagonal equation. If we assume an accessory square root, we can ensure that .^4 = in Kronecker's resolvent. The corresponding diagonal equation is then essentially identical with the equation of the xCs, which we just considered in § 4. The curve of the m's was of the twelfth order, or split up into two half-regular curves of the sixth order. Our general proposition would, therefore, for this also lead to an auxiliary equation of the sixth degree after adjunction of the square root of the discriminant. Nevertheless, as has just been stated, a single additional square root is sufficient. § 7. Geometrical Aspect of the Formation of Resolvents. The algebraical principles of the construction of resolvents have already been thoroughly explained in I. 4 for arbitrary algebraical equations. Their specification for equations of the fifth degree needs in itself no corollary. If we return to this here, it is only to give a new application to our former remarks. We premise, in the first place, that we shall only introduce such rational functions of the a.''s : assumes in consequence of our permutations, are shown to be invariants for every case, and we can therefore, by deno- ting the ^'s as homogeneous co-ordinates, interpret the forma- tion of the resolvents in a geometrical way. This is a limita- tion which we make merely in favour of our geometrical inter- pretation ; it has no deeper significance, and can hereafter be dispensed with. Corresponding to the basis of analytical geometry, two pos- sibilities now occur at the outset for the interpretation. Either we consider the introduction of the ^'s as a mere change of the system of co-ordinates, or, in Pliicker's sense, as a change of the elements of space. In the first case, the ^'s appear directly as homogeneous, and in general curved co-ordinates of a point, between which (■".—4) identities necessarily exist. In the second case, the ^'s are primarily independent magnitudes, which we denote as the co-ordinates of any geometrical figure. The choice of this figure is only restricted by the condition that its co-ordinates, on the introduction of the 120 or GO coUinea- tions of space which we are considering, experience just the same permutations as the ^V ^2' •'3' ■'4' putting (19) p,,=X,Y,-r,X„ we have in the first place : (20) Pit^ -Pki, by means of which the twelve different ^^^.'s which occur are reduced to six linearly independent ones, for which we choose, say, the following : (21) Pn, Pii, Pw Pat' P4.2' Pis- Between these there then exists in addition the following easily- proved identity : (22) P = P12P34 +P13P42 +PuP2x = 0- Two lines intersect when a bilinear relation obtains among their co-ordinates, which we can denote briefly as follows : (23) 2^-£=^- The summation has here to extend over the six combinations (21). This is clearly not the general linear equation for the Pji's, for the pn^'s are also subject to an identity of the form (22). Understanding by a^^ arbitrary magnitudes, and keeping to the * Republished 1878. 194 INTRODUCTION OF GEOMETRY. table (21), we will write the general equation in question in the following form : (24) S«-£-»^ The assemblage of the straight lines which satisfy an equation of this kind is what Pliicker has called a linear complex, while Mobius in 1833 has discussed it more completely. AVe will not here concern ourselves with the geometrical properties of the linear complex any further. We will only add that we shall denote the coefficients «,j. as co-ordinates of the linear complex, where we may introduce, if we please, in accordance with formula (20), beside the «,t's other symbols Aj.;: (25) a,,= -a,,. If: (26) ai^rtj^ + flija^j + nu«23 = 0. we can replace the «a.'s by the Pn^s of the formula (23), the complex is then a special one, and consists evidently of all straight lines which intersect the fixed line p'. If we combine by addition two special complexes p', p", and so construct : we have, apart from particular cases, a general complex. Every gcnercd complex can he obtained hy adding together six given special complexes with the help of proper multipliers; only the special complexes must be linearly independent, i.e., they must not satisfy by their co-ordinates the same linear homogeneous equation. In this sense, in particular, the six straight lines are available which form the edges of a tetrahedron. So much for the usual conventions of the line-geometry. If we now replace the co-ordinate tetrahedron by a pentahedron, the only modification is this : that the number of the co-ordinates appears changed, but, to meet these, new equations of condition occur. First as regards the point co-ordinates, we have for X, Y, now, just as before : X(|, Xj, X.,, X^, X^ ; Tg, Fj, 1\, Fj, Y^, with SX=0, Sy=0. But then ire have ticeniy determinants : INTRODUCTION OF GEOMETRY. T95 (27) p,, = X,Y,-Y,X, to distingtiish. We have again, of course : (28) Vo:= - 1'" : but besides this, evidently : (29) 2i'a = 0> or also 2l^.. = 0, where the summation extends over those four values of i and h respectively, which are different from the corresponding h and i. Besides this, there exists the quadratic relation (22) in addition to the others which proceed from it by means of (28) and (29). Again, we can also speak of co-ordinates of the linear complex. There are twenty magnitudes «jj. which, while satisfying the linear relations (28), (29), are otherwise unrestricted variables. What was said with regard to the composition of general linear complexes out of special complexes remains valid. All these matters are so simple that we can now break off any further consideration of them. § 9. A Resolvent of the Twentieth Degree of Equations OF the Fifth Degree. Let us go back again to the considerations of § 7. We wished to consider those equations on which the pentahedral co-ordinates of a straight line in space depend. We can evidently, instead of these equations, at once take into consideration the more general ones by which the co-ordinates of an arbitrary linear complex are determined. We thus obtain in general equa- tions of the twentieth degree whose roots fl;^, in conformity to the formulae (28), (29), are connected by the following linear relations : (30) a«=-n,, Sa,, = 0, Sa,, = 0. A certain similarity between these equations and the Jacobian equations of the sixth degree (in so far as we regard the latter, as Herr Kronecker does, as equations of the 12th degree for the n/s's) is from the very first unmistakable ; we shall learn later on (in the fifth chapter) the intimate connection that actually exists in this respect. 196 INTRODUCTION OF GEOMETRY. Our business now is to make the magnitudes a^. equal to proper functions of the xb, and to turn equations of the 20th degree into resolvents of the equation of the fifth degree. The plan is, as we expressed it in § 7, to connect the linear complex (whose co-ordinates are ajj.) with the point jc as a covariant. We effect this in a simple manner if we rely on § 5. We have there constructed the «'", x^^\ a;"', a;'*', as the simplest covariant points of the point x ; we shall obtain the simplest covariant straight lines if we consider the lines which join these points. The co-ordinates p^jc of this line : (31) p'^,'-=x'y^^-a^r4' are linearly independent, for we have to do with the six edges of a tetrahedron. Therefore we shall obtain the most general values of ftji by combining these j?ji's with the help of proper multipliers : (32) a,, = 2c'-'"-y.r- Here the c^-'^'s are to be introduced as symmetric or as two- valued functions of the x's, according as we choose to consider all the permutations of the x's, or only the positive permutations thereof; but otherwise they are to be chosen so that the law of homogeneity that we accorded is satisfied. § 10. Theory of the Surface of the Second Degree. I conclude the present chapter with some remarks on the iTistitution of parameters for the linear generators on surfaces of the second degree. The parameters in question are linear multipartite functions of the projective point co-ordinates.* We obtain them most simply by bringing the equation of the surface (as it is possible to do in an infinite number of ways) into the following form : (33) X^X^ + X,X^ = 0. If we put then, firstly, in correspondence with this equation : * The introduction of this parameter is an equivalent, geometrically speaking, of the projective generation of the two families of ruled lines on the surface, which, for example, Steiner makes the basis of his considerations. K 'X, ' ^^3 INTRODUCTION OF GEOMETRY. 197 (34) ^i = -^3^x secondly : (35) X remains constant when we move along a generator of one kind, which might be called the first, while /u. remains con- stant when we proceed along a generator of the second kind. Therefore X, fi, are two numbers which are characteristic of the individual generators of the first or second kind, i.e., they are parameters which can be used to distinguish the generators. Here we observe that each of the formulae (34), (35), embraces two equations. We may therefore, without changing the mean- ing of \, fi, generalise their definition somewhat. For \, for example, by combining the two equations (34) with the help of arbitrary magnitudes p and cr, we can write : (36) X=Zi^i±^3_ We succeed in making numerator and denominator of A. vanish together for an arbitrarily chosen generator of the second kind: e 1/.= . P The generator chosen in this manner shall be called the basis of the introduction of \. We will now first consider the behaviour of X, fi, with respect to such space coUineations as transform our surface into itself.* The coUineations in question arrange themselves, as is known, into two kinds, according to their behaviour with regard to the generators of the surface : either they transform each of the two systems of generators into itself, or they interchange the two systems. In the first case, to each generator corresponds, in virtue of the presupposed collineation, one, and only one, gene- rator X'; and conversely, in the same way, to every /i corre- * Consult, say, Bd. ix. of the Math. Ann., p. 188, &c. The theorems intro- duced in the text are also often used in other departments of modern research. A^ more thorough proof would take us too far. 198 INTRODUCTION OF GEOMETRY. sponds one /a'. Therefore we have, on function-theory principles, corresponding to such a collineation, formulas of the following kind necessarily appertaining : aX + i , a'fi + V (37) cX + d' c'/J- + d' In other cases A,' by analogy will be a linear function of ^ ; fJ-' such a function of X. I do not stay to show that these propo- sitions can also be reversed, and that therefore a corresponding space collineation is obtained if the formulce (37) [or the corre- sponding ones in which X and ^ are interchanged] are written down quite arbitrarily. We remark, moreover, that the X, fi a furnish a determination of co-orclinates for the points on our surface* In fact, at every point one generator of the first kind and one of the second intersect, whose X, /u, we can transfer to the point. It is here to the purpose to replace X by X^ : Xj. M ^1 Mi • A'2' *° make them homogeneous. An algebraical equation : (38) f{\,K; i-vt^2) = Q, homogeneous and of degree Z in > j, X,, of degree tn in fi^, fi^, then expresses a curve of the (l + nif^ order lying on the surface, which intersects a generator of the first kind 7)i-times and one of the second kind ^-times. We can now combine (34), (35), in the following manner : (39) Xj . ^2 : X.^ : X^ = Xj^j • - Xj'ij : X^.a^ : X^a^. Introducing these values of the JT's into the equation of a surface of the 71"^ order : (40) F{A\,A\,JC„X,) = 0, we recognise that oar surface of the second degree is intersected by (40) in a curve which, written in the form (38), is of the w* degree both in the X's and ^'s. Conversely, too, by means of formula (39), every curve (38) which is of equal degree in the * See Pluoker in Crelle's Journal, vol. xxxvL (1847). The discussion of the enrves (38) was undertaken in a systematic manner almost simultaneously by Mr. Cayley and by Chaslea (1861) ; see Phil. Mag., vol. xxii., also Coniptes Kendus, vol. liii. INTRODUCTION OF GEOMETRY. 199 X, ^'s can be represented as the complete intersection of the surface of the second degree with an accessory surface (40).* We determine, finally, the line co-ordinates of the generator X, /i, retaining the tetrahedron as laid down in (33). Putting first /ij = 0, then fio = in (39), we obtain for two points lying on the generator \ : X^:X.,:JC3:X^ = 0: iX^ : K Y-,: y,': 7-3: Y^ = X^:-A^:0 : o" respectively. Hence we calculate by (19) for the 2JiA;'s which belong to them the following relative values : (41) I'l, = 0, Pn = >.i-, Pii = \K, Psi = 0, i^i.2 = X^a, j,,^ = - Xj \,. Analogously we get for the ^-generator : f 42) 2h2 = - f^i-' Pis = 0. Pu = ^iM2> i'34 = .<*/. 2\t = Oj i'23 = f^-^f-r We now assume that the equation of a linear complex is intro- duced, which runs as follows : By inserting herein the expressions (41), (42), we obtain the two following quadratic equations : (43) ^42^' + (-4.3 - ^14) ^>-2 + ^13>-2'- = 0. (44) - ^34."l' + (^•.;3 + ^14) .V^2 + ^1.-.-'= <^- Hence : 1)1 general to a linear comijlcx hcloufj iico, and unlij iicu^ ijenc- ratms 0/ each system. But it may happen that one or other of these equations * I might perh.'vps add one remark, which is not immediately connected with the text, but rather reverts to the developments of the first part, viz., this : that Kieniann's interpretation of x + i^ on the sphere can be applied as a special case of the determination of X, p., co-ordinates spoken of iu the text, namely, since all the linear generators on the sphere are imaginary, two conjugate imaginary generators intersect in every real point thereof. If we now introduce X, yu, properlv, and call the X, which belongs to a real point on the sphere, x+ii), then the corresponding fi. will be x-iij. For fiximj the real point, therefore, it suffices to give only one value, x + iy, and this is just the method of lliemauu, as I cannot further explain here. if. Math. Annalen, lid. ix. p. 1S9 (1S75). 200 INTRODUCTION OF GEOMETRY. vanishes identically. This gives three linear conditions for the A ij's, so that three of these still remain arbitrary. Hence : The generators of the first and second kind on our surface belong each to a threefold linear family of linear complexes.* I must pass over the actual establishment of the equations of these families. * Cf. throughout Pliicker's " Nene Geometrie des Eaumes," &c. ( 20I ) CHAPTER III. THE PRINCIPAL EQUATIONS OF THE FIFTH DEGREE. § 1. Notation — The FuNDAME>fTAL Lemma. The new chapter which we now begin is to form in every respect the centre of our developments. We treat of the prin- cipal equations of the fifth degree and their simple relations to the ikosahedron. Here we borrow from what precedes, espe- cially from the Bring transformation, the one fundamental idea of considering the rectilineal generators of the principal surface. T denote here, as I did there, the principal equation of the fifth degree as follows : (1) y^ + 5a,f + 5Pi/+y = 0, where the factors 5 for a and /3 respectively are applied for the sake of convenience. I will also communicate at the outset the value of the discriminant. Using the somewhat long formula which we frequently find* given for the discriminant of the general equation of the fifth degree, we have for (1) : (2) n(y,-2/,)2 = 3125v=, where V' is put for brevity in place of the following expres- sion : (3) v== lOSaSj- _ 135a4/32 + gOa^^j-S _ 320a/33y + 25C/35 + yi. We now at once annex the developments just given (in the concluding paragraph of the preceding chapter) by supposing the two different generators of the principal surface to be de- noted by parameters \, fi. Let : * Cf. e.g., Fad di Bruno, edited by Walta; " Enleitung in die Theorie der binaren Formen " (Leipzig, 1881), p. 317. 202 THE PRINCIPAL EQUATIONS OF (4) Vq, Vi, 2/2, 2/3, 2/4 be the roots of (1) in a definite order. We then suppose those 60 generators X and 60 generators /j, constructed which contain one of the 60 points of the principal surface, whose co-ordinates proceed from (4) by an even permutation of the y's. The \ /I's are, as we know, linear fractional functions of the y's ; the 60 values of X or /i in question therefore depend on an equation of the 60th degree, which is a rational resolvent of our principal equation, and the coefficients of which are accordingly rational functions of the a, /3, 7, y. Noiu I assert — and here we have the particular lemma required for our further developments — that our resolvents of the QOth degree, for an appropriate intro- duction of the X, /i's, are neeessarily ikosahedral equations, and therefore loill admit of being written ivithout more ado : where Z^, Z^, alone depend on a, /S, 7, y. The proof presents itself immediately on the grounds of our previous data. We have just divided the collineations of space which transform a surface of the second degree into itself into two parts, according as they transform the individual system of generators into itself, or interchange it with the other system. Now the principal surface of the second degree passes into itself for the 120 collineations of space which correspond to the permutations of the ys. We will at first leave undetermined how the systems of generators of the surface behave towards the totality of these collineations. If all the collineations were not to transform the individual system of generators into itself, at all events half of them would necessarily do so. This half of our collineations must here necessarily form a group j)er se, and indeed a self-conjugate group in the main group ; it can therefore only consist of the even collineations. Hence, in any case — and this is a first result — the 60 even collineations have the property of transforming each of the tioo systems of generators of the principcd surface into itself. We now recall that, in accord- ance with the formula just given in (37) [II. 2, § 10], the para- meter X, as also the parameter ft, experiences on its part a linear transformation for each collineation of this kind. The 60 values THE FIFTH DEGREE. 203 \, which satisfy our resolvent of the bOth degree, therefore depend on each other as linear functions with constant coefficients (and similarly the corresponding vahies of /j,), or the equations for \ aiul fi are transformed respectively into themselves hy a group 0/ 60 linear substitutions. But hence the accuracy of our asser- tion follows immediately in virtue of the developments of I. 5, § 2, as soon as we add that the group of the linear transforma- tions which \ or fj, experiences is holohedrically isomorphous with the group of even permutations of the y'a. The unknowns \, fi, which occur in the canonical forms (5), are here proper linear functions of the original parameters denoted by these letters ; we will call them the normal parainieters, not forgetting, however, that they can be chosen in sixty different ways in correspondence with the 60 linear transformations by which each of the equations (5) is transformed into itself Having thus proved our primary assertion, we can go a step farther in the same direction. I say first, again taking up the question just mooted, that for each uneven coUineation the two systems of generators of the principal surface are necessarily interchanged, namely, if the individual system were trans- formed into itself for the whole of the 120 collineations, a group of 120 linear substitutions of a variable would be given, on the grounds of the formula (37) just cited, which would be holo- hedrically isomorphous with the group of 120 permutations of five things, which, however, by I. 5, § 2, is impossible. If, therefore, we have represented \ (the parameter of the gene- rators of the first kind) in any way as a fractional linear function of the y's, we obtain a parameter fi of the generators of the second kind by subjecting the y's occurring in X to any uneven permutation. In particidar, loe obtain the sixty normal values of fj, if we apply to one of the normal values of X the wJwle of the uneven permutations of the ys. For these uneven permutations the coefficients a, /3, 7, of course remain unaltered, while V changes its sign. The magnitudes Z^, Z^, occurring in the equations (5), only differ, therefore, in the sign of V- ^^ can give this theorem another application by introducing the sixty points y, whose co-ordinates are derived from the scheme (4) by uneven permutations of the y's. We have, namely, for the representation of the generators of the first ami second kind 204 THE PRINCIPAL EQUATIONS OF which pass through these points, the following equations respec- tively : § 2. Determination of the Appropriate Parameter X. The formulae which we will now establish for the normal \ are in themselves peculiarly simple and easy to verify. If I nevertheless devote some space to deducing it, it is because I again wish to derive each individual result from reflexions which involve no computation. As the generative operations of the ikosahedral group we have previously (I. 2, § 6) found the two following : (7) -l (e-ji^z+Jf-^^) We saw, further (I. 4, § 10), that the octahedral forms t^ are permuted as follows with respect to these substitutions : ^ ' i T ■ f ' = f f ' = t t' —t 1 ' — 1 t ' = f ( ± . 'o 'OJ 'l '2' '2~'l> '3 ~ '4' '•! '3' The same formulae of permutation hold good for the roots of the several resolvents of the fifth degree for the ikosahedron * I first gave the reasoning developed in the text (as well as the corresponding formula? of the two following paragraphs) in two communications to the Erlanger SociBtiit on November 13, 1876, and January 15, 1877 ["Weitere Mittheilungen liber das Ikosaeder I, II "]. I will now append, besides, the case of the equations of the third and fourth degrees for comparison. Let us denote the three roots x of an equation of the third degree having Sx = 0, in accordance with what has gone before, on a straight line. Let us then denote an arbitrary point of this straight line in the usual way by a parameter \ ; then X, for the whole of the six permu- tations of the xs, experiences linear substitutions of the dihedral type, and satisfies, when properly prepared, a dihedral equation of the sixth degree. For the equations of the fourth degree we transfer the geometrical represen- tation to the plane, and add to the condition 2{x) = the second one 2ar = 0, confining ourselves therefore to "principal equations." We again represent, in the usual manner, by a parameter \ the points of the selected conic. This para- meter then undergoes linear substitutions for the whole of the twenty-four permu- tations of the x's, and therefore satisfies, when properly prepared, an octahedral erjualion (or, after adjunction of the square root of the discriminant of the equa- tion of the fourth degree, a telrahedral equation). THE FIFTH DEGREE. 205 which we there established (I. 4), and in particular — a point to which we shall soon return — for the roots of the principal re- solvent. We shall want to arrange our new formulae so that they fit in with those there given as closely as possible. We shall therefore so choose the normal \ from among the sixty values of the parameter which come under consideration, that it undergoes exactly the substitutions (7), if ive subject the y's to the tioo fcr- Tnutations indicated by (8). The value of \ is fixed hereby, but not so its form as a func- tion of the y'a. First, we have yet to decide which generator of the second kind we will make the basis of the introduction of \ in the sense previously explained (il. 2, § 10). Secondly, we can modify numerator and denominator of X by addition of arbitrary multiples of Xy (which is identically = 0). In both respects we will make definite conventions. For each linear substitution of A, or /i two values of the vari- able, i.e., two generators of the first or second kind respectively remain fixed. We consider now in particular the operation S, and make the basis of the introduction of X one of the two generators of the second kind which remain fixed under its 7} action. Let X, on this supposition, = - , where p and q denote two linear functions of the y's. On efiecting in p, q, that per- mutation of the y's which is likewise indicated by S, p', q' arise. Here p' = 0, q' = 0, have by hypothesis the same straight line in common as p = 0, q = 0; therefore for any y : p' = ap+ hq + m ■ 2 p/, q =cp + dq + n- 2//, but ^, = >-' is, in accordance with formula (7), to be equal to eX for all points of the principal surface ; and the points of the principal surface are not distinguished from the other points of space by any linear relation among the co-ordinates. Hence the foregoing equations are necessarily transformed into the more simple : p = ul ■ p + m ■ ~!/, q = d ■ q + n ■ -g, where d, m, n, are primarily unknown, We can modify these equations as follows : 2o6 THE PRINCIPAL EQUATIONS OF m „ J / m _ \ We shall now be able, without afifecting the equation ?>• = , to denote the expressions P + ~j -^ • 2?/, q+ , ^ ■ -y, which occur, in a concise manner, by p, q. Then we have simply : (9) lP=^^^P, The result of this reflexion is thus as follows : we can put, and this in tivo loaj/s (since one of two generators of the second kind had to be chosen), our X=- in such ivise that, after application of the permutation S to the y's, the equation (9) is identically true. Now, however, it is known (and, moreover, easy to prove) that, for the permutation S of any magnitudes y, no other linear functions of the i/'s alter only by a constant factor save multiples of the expressions of Lagrange : (Px = Va + ^'/i + ^"^2 + ^^1/3 + '■'.'/4' Pi = 2/o + «-.'/l + '^2 + % + ''^4' Ih = ya+ ^I'/i + fZ/o + «Vs + ^^Vv Pi = Vo + ^*!/l + ^^IJ2 + ^-y-i + «.'/4' with which Sy, as an expression which remains entirely un- altered, would also be associated were it not in our case identically zero. As regards the changes of the 2h'^ for the permutation S, p'^ = e'"' ■ p,,. Hence the only three expressions for X which satisfy the relations (9) are the following : (II) x=,.g, -v|, x=v». of these expressions, the first and the third are available, but the second must be rejected. It can be shown, namely, that the line of intersection of^j = andp2 = 0, as also that of ^3 = and 2^4 = 0, belong, in fact, to the principal surface, but not the straight line p.2=0, ^3 = 0. This is best proved by introducing the p^'s in place of the yjs in the equation of the principal sur- face. We have from (9), on joining thereto Sy = : THE FIFTH DEGREE. 207 (12) 52/, = e^-jij + t^''p., + i^"}-)^ + ep^, and therefore : so that the equation of the principal surface, relatively to the co-ordinate system of Lagrange, will be the following : (13) PiPi+P22h = ^; whence the accuracy of our statement follows directly. From (13) it follows, further, that : jP2 Pi' if, therefore, we put, corresponding to the first formida (11), \ = Cj • — , ^ue must also put it = — Cj • - . It will now only remain to determine the factor c which occurs here. We construct X' by submitting the y's to the permuta- tion T, according to formula (8), and then inserting it in the corresponding formula (7). The equation thus arising cannot be an identity, because the generator of the second kind, which we used in establishing the X, does not remain fixed under the action of T. It must, however, be a valid equation if we take account of the relations Sy=0, Sy^ = 0- We obtain, on com- paring the proper terms on both sides, the value — 1 for Cj. Hence our normal X. is determinate : (14) x=-^^=Ps, P2 Pi in exact agreement with the value which we had adopted for the parameter \ in the concluding paragraph of the preceding chapter (formula (34)).* * If we wish to establish in a similar manner for the equations of the third and fourth degrees, which we just mentioned, the roots of the dihedral equation and octahedral equation respectively, we obtain accordingly : X o + ax T, + a^ V^( ro + ii^+ i-x.2 + Pxs) ^ " (xf , + i'^Xi + i*x. + iH s) ~Xo + a-Xi + ax-i ~ + (a^o + rxi + i-'x^ + i^X3)~ ^2 (xo + i'xi + {"x-. + 1^1-3) where a':=i''=l. so that, therefore, the quotients of the expressions of Lagrange are here also introduced. 2o8 THE PRINCIPAL EQUATIONS OF § 3. Determination of the Parameter /j,. The normal parameter /j,, which we had in view for the generators of the second kind, was to proceed from the para- meter X, by means of an uneven permutation of the ys. We satisfy this requirement if we take, again in agreement with the concluding paragraph of the preceding chapter (formula (35)) : (15) ^=-^2=^1. Pi Ps In fact, this value results from (14) if we replace y^, y^, y^. y^, y^ ^J 1/o' Vs' Vv 2/4- Vi respectively, and therefore permute y^, y^, y^, y^ cyclically. But now we can evidently deduce the formula (15) from (14) in yet another way, viz., by replacing in (14) e by e^ through- out. This change is then, of course, carried over to the sub- stitutions S, T (7), and the ikosahedral substitutions arising from them. The suhstitutions, therefore, which jj, and X undergo for the even permutations of the y's, thongh by no means identical individually, are so, at any rate, in their totality ; or rather we derive the one set from the others by changing e into e^ throughout, a theorem which is fundamental in what follows. In agreement with this we obtain, on applying to the fi the operation men- tioned, not \ again, say, but — -. This is that value which A arises from X in virtue of the ikosahedral substitution denoted previously by U. To it corresponds the simultaneous inter- change of 7/j with y^, and of y^ with y^. We will further adopt the formula (39) of II. 2, § 10. In virtue of this we now have, on r.eplacing X by Xj : X^, /i by ^1 : /i, : (16) X>i-P2- Ih ■ I'i = '^1^1 : - '^2."i : ^f^i ■ ^■2"2' or, on introducing a proportion-factor p : (17) PI/., = ^' ■ \f^i - f'" ■ Xa"*! + «^'' • '^i'"2 + *■' • '^2'^2- THE FIFTH DEGREE. 209 § 4. The Principal Resolvent of the Ikosahedral Equation. Having found the normal parameters \, /x, for any principal equation of the fifth degree, we will apply our formulae in par- ticular to the principal resolvent of the fifth degree, which we previously constructed, IV. 1, § 12, by supposing any ikosa- hedral equation : ^l^-* 1728/M.,,.,)-^ to be given. We obtain in this manner a peculiarly simple result, which is of proportionately great importance for the further progress of our development. The principal resolvent was defined by the formulas : (19) Yy = m .v^ + n . u^Vy, where (20) «. = — ji — , 1^= j^^, understanding by /, ff, T, the ground-forms of the ikosahedron, by ty, TFj,, the oft-mentioned forms of the sixth and eighth degrees. Let us now consider that we can write Wy and <„ W^, in the following manner : (21) IF, = (e*% - ^•'z.^ ( - 2i' + T.-i^x,"') (22) i, W, = (£*-«i - £3-22) ( - 26zi>%3 + 39zi V + ^,^13) + (€i-z\ + fz,) ( - Zjis + 39zi%5 + 26z,W'>). Hence Y, in formula (19) assumes the following form : (23) r, = {^"z, - ^^"z.,) R + (e-'z, + €"7.^) S, where B, S, are linear functions of m, n. The expressions of Lagrange (which we also here denote by capital letters) become therefore : (■ P, = 5zj . B, P3 = 5r.^ . S, ^'^'^^ \P,= -5z,.Ii, P, = r,,:,..S. Hence we get simply : 2 10 THE PRINCIPAL EQUATIONS OF (25) x=^, ^=^ We have therefore at the outset : Th.e parameter \ is identical with the tmJcnoivn z^ : z^ of the original ikosahedral eqiuition, or expressed geometrically : TJie point (26) y^ = m. Vy{X) + n . u^(X) . v^(X) lies on a generator of the first kind, whatever m and n may denote. If we here consider X as a variable magnitude, the point y^, as we saw in the fourth paragraph of the preceding chapter, tra- verses a half-regular rational curve, which is in general of the 38th order. For the proof of this, we had replaced the formula (26) by the following (a proportion -factor p being introduced) : (27) ',y.^m.WA\,\)-T(\,\) + \2n. t, (X„ >.,) . {W, (X„ X^) ./2 (Xj, Xj). We now recognise, first, as was cursorily remarked, the reason (geometrically speaking), why the order of the curve thus ob- tained can sink to 14 for m = 0, and to 8 for w = 0. It is be- cause, in the first case, the aggregate of the 12 generators of the first kind /(Xj, Xg) — ^j counted twice, is separated from the general curve of the 38th order; in the second case the aggre- gate T(\, \^ = 0, counted once. But we have now, besides, the following theorems for our curves (27). We find that our curves meet the generators of the first kind only once, and therefore the generators of the second kind 37 times. In fact, we have for every generator X by (27) only one point of the curve. We find, moreover, that through every point of a generator X only one curve (27) passes, so that the principal surface is covered hy the family of curves (27) exactly once. The individual points of the generator X, viz., are given by the corresponding fj,, which deter- mines the generator of the second kind which passes through the point. But if we suppose X, /i, in (25) to be known, the corresponding m : n is computed linearly. We append to this two further remarks which will be useful later on. First, as regards m and n, we can compute these linearly from the y„'s previously given in accordance with formula (26), not merely relatively, but determining their absolute values. These formulae are not altered if we permute THE FIFTH DEGREE. 211 the 2/y's evenly in any way. For by the agency of the ikosa- hedral substitutions of \, the w„(\)'s, i?^(X)'s occurring on the right-hand side always undergo the same even permutations as the yjs placed on the left side. Tlie m, n, there/ore depend rationally on the yjs in such wise that they remain unaltered for even permutations of the yjs; or, to express it otherwise, the m, n, are capable of rational representation as fxinctions of the given magnitudes a, )8, 7, V- ^^ consider further the relation between \, /t, which is famished by the formula (25). If we subject X to any of the ikosahedral substitutions, the fi, inas- much as it depends on the corresponding 3^/s (exactly in the way we saw in the preceding paragraph), undergoes other ikosahedral substitutions which proceed from the given ones by changing e into e^. Following the terminology which was introduced in this connection by Herr Gordan, we will describe the changes of ft as contragredient to the changes of X. The formuloB (25) provide us vnth infinitely many rational functions ofK which, in this seTise, are contragrediently related to X.* § 5. Solution of the Principal Equations or the Fifth Degree. We have already, in § 1, 2, given the means of reducing the solution of the principal equations of the fifth degree to an ikosahedral equation : by determining X as a function of the y's. If we now wish to express conversely the yjs by means of the individual root X, we can evidently employ the equation (26). I will now write it so that m, n, are provided with an index 1, so that the con- * The theorems proved in the text, as well as the principles for tlie solution of the principal equations of the fifth degree to be developed immediately, were brought before the Erlangen Society by Herr Gordan and myself simultaneously on the 21st of May 1877. Herr Gordan there started from essentially different points of view from those to which we afterwards return. My own exposition, too, was in some measure different from that now given in the text, and in many respects less simple. Cf. here throughout my comprehensive memoir, " Weitere Untersuchungen iiber das Ikosaeder," in Bd. xii. of the Mathematische Annalen (August 1877). 2 12 THE PRINCIPAL EQUATIONS OF nection of our formula with the ikosahedral equation (28) may be evident. We have then : (29) y, = ?«i • i\{>-) + J?j • «^(>-) • v^{x) ; when we afterwards consider fi instead of X, -^j, ?ftj, Mj, will have to be simultaneously transformed into Z^, m^-, w,. In order that the solution of the principal equation by the help of the ikosahedral equation may be complete, we have evidently only to further determine the Zj^, ?)ij, w^, as rational functions of the magnitudes a, /3, 7, V) previously given. We shall see later on how the calculation thus required can be carried out a p^Hori. In the meantime, let us follow a much more elementary method. We have in I. 4, § 12, explicitly computed the principal resolvent of the ikosahedral equation by considering Z, m, n, as arbitrary magnitudes, and in | 14 have given the corresponding square root of the discriminant. It now follows from the considerations of the preceding paragraph that every principal equation of the fifth degree, after a fixed value has been determined for V) admits of being put, in one way only, into the form of the principal resolvent. We shall there/we he able to determine Z-y, v\, «j, in a rational manner iy simply comparing the coefficients of the general principal resolvent and the square root of its discriminant with the coefficients a, /3, 7, of the given principal equation (1) an^ the adjoined value of the corresponding V- We will here always define V) as we did in I. 4, § 14, in the following way : (3O) 25V5.v = 7T( {y^ - Vv which is reconcilable with the formulas (2) and (3) of the present chapter. Comparing now, first, only the two sets of coefficients, we obtain : Z- a. = ^w? + Vlmhi-v — ■, _y -, Z-B , . 6m^n^+imn^ 3n* (31) ^ -^/--im*+ ^_^ -^4(13^)-^' Zy 4:0m^u^ I5mn* + iifi THE FIFTH DEGREE. 213 I have here at the outset written Z, m, n, instead of Z^, wij, «i, because Z2, m^, n^, satisfy these equations equally well. The further computation now takes the following form.* From the first of the equations (31) we obtain : On the other hand we form : 1-2 ,„ „^o 1 We need only introduce here the value (32) of ^j — y, in order to obtain for m a quadratic equation. If we rearrange this by multiplying up by the denominator, we have : (34) Um"- (a* - y83 + a/3y) - tn {lla^lS + ifi^y - af) o + ^ (64a2/32 - STa^y - (iy") = 0. On solving this we find : where V" exactly agrees with (3) (as we may verify), and the sign ± remains for the time, of course, undetermined. With this value of m the other unknowns are at once determined too. First, as regards the value of Z, it is sufficient to introduce the value of T — — from (32) into the first of the equations (33) ; we thus find : m) 7- (48« m8-12/3m-y)3 * I borrow the process of elimination used in the text from a lecture of Gordan's in the winter 1880-81. HerrKiepert has also similarly employed the comparison with the principal resolvent ("Auflbsung der Gleichungen fiinften Grades") in the Gottinger Nachrichten of July 17, 1878, or Borchardt's .Journal, t 87 (1879). 214 THE PRINCIPAL EQUATIONS OF We obtain n in a corresponding manner if we write the first of the equations (31) as follows : (l2m^ + ^-^\ n = aZ-8m?- 6m • ■^^, and now consider 711 and Z as known. The final formula is : (37) n=- ^ 96am3 + 72/3mg + 6ym - 12tt^Z liiam' + 12jSm + 7 In order now to determine the sign of V in (35), and there- fore in (36) and (37), in a way corresponding to the priority of the X and the notation m^, Jij, Zj, let us compare (30) with the difference-product of the principal resolvent previously noted. It here suffices to consider a special case. We take, say, m = 1, 71 = 0, in the general principal resolvent, and therefore have, in consequence of the formula (31) : 8 „ 12 144 " = z' ^=-z' ^ = ^- At the same time we obtain by I. 4, § 14 : J7< and thus by formula (30) : ^ 12^(1 -Z) Now, by formula (35), the m becomes in this case : -11 . 28-3. 123.Z±4. 123 (Z-1) 2'-'-7.12^Z ' therefore if, as we assumed, m is to be equal to 1, we have to apply the lower sign in (35). Thus we have in general : _ (lla3,3+2g^y-«y^)-aV and hence by (36), (37), the Z^ and Wj. The corresponding values of m^, Z^, n^, proceed from this by reversing the sign of V throujjhout. THE FIFTH DEGREE. 215 § 6. Gordan's Peocess. The method just developed for the computation of m-^, n^, Z^, has the advantage of working throughout on elementary lines, and with the use of results previously deduced. It cannot, however, be denied that a certain amount of skill, though of a very simple kind, is required to introduce the right combina- tions of the equations (31), and that, therefore, this method does not fit in well with the mode of exposition which we have otherwise maintained, in which we have endeavoured to always have an insight a priori into the results of calculations, so far as regards their nature. I will therefore briefly go into the nature of the computation originally given by ITerr Gordan, and this the more readily because certain other aspects are connected with it which are of use for our main conception of the problem of solution.* Let us first make clear the difiicnlties which oppose a direct computation of the magnitudes 2'j, mj, n-^. We had, for example, the defining equation : 1 1728/5 (X)' 1728/5 (X)' where we may substitute for X the one value : .Pi P2 Then we have in Z^ a rational function of the five roots 2/0 • ■ • ^4 before ns which remains unaltered for all even permutations of the y's. But now this latter only occurs because the y's are connected by the equations of condition Sy = 0, 2y^ = ; it does not occur if we consider the y's as arbitrarily variable magni- tudes. To express it otherwise, Zj^ is for the even permuta- tions of the y's actually, though not formally, invariant. Now all rules which we meet with in the usual expositions on the computation of symmetric functions, &c., relate to functions of formal symmetry; these rules are, therefore, not immediately available for our purpose. * Cf,, besides the note just mentioned, a communication of Gordan's to the Naturforscherversammlung at Miinchen (Sept. 1877), as well as the larger memoir, "Ueber die Auflosung der Gleiohungen vom fiinften Grade," in Bd. xiii. of Math. Annalen (January 1878). 2i6 THE PRINCIPAL EQUATIONS OF Herr Gordan surmounts this difficulty by satisfying the equations of condition Sy = 0, Sy^ = 0, in a general manner by functions of independent magnitudes. He then has henceforth to do altogether with functions of independent variables, and can establish for them an algorithm which is analogous to a certain extent to the process already mentioned relating to symmetric functions. The dependent variables from which Herr Gordan starts are essentially none other than the homogeneous parameters \j, Xji and fi^, fi^. We have above already expressed the ratios of the p^'s, and, on the other hand, the ratios of the yjs in terms of these magnitudes [formulae (16), (17)]. Herr Gordan renders the formulae in question concise by supposing the absolute values of the \, /Li's determined appropriately, and writing accordingly as follows : — (39) p^ = 5\fi^, P2= - ^\<^V Pi = 5^l'*2> Pi = ^\l^i^ whereupon y^ becomes equal to the following expression : (40) 2/. = e*" ■ \!J'i - e^" . >'2'"l + ^^' ■ \f^2 + ^' ■ \'^-2- Before going further in the description of Gordan's process, we will express all the magnitudes, given and required, by the \, /t's thus introduced. I first bring together the formulae for the coefficients a, /3, 7, of the proposed equation of the fifth degree and the corresponding V. We have : (41) a = - Y5 = - '^Wl'-z - \^\l'i^ - WW + V^l^2^ (42) /3 = - lo = - '■l"''!''^' + ^1' Vl' + S\'\Wl^2' - >-l>-2V2^ + V/hV2' (43) y=-^=- >./(^i^ + ,,/) + \0\%^,%^ - 10X/X,ViMo* - \o\^\^^,% - lOXjXjViV + \'W - H')^ 25^y5 (44) THE FIFTH DEGREE. 217 + Xj9x,(25/ii V - 50/^1^2^) + W^i - 50/^1 V/ - 25AtiV2') + '^i'^2'( - 75,«i«^2* + 25//.i"2«) + \^W - 25/*iV2 - 1W^2) + Xi7x/( - 50/^1 V2 - 150^1 V) + '^i^V( + ISO.","/*,* - 50/XiM2«) + Xj6x24(l50^iV/ + 75aii V) + \'V( + 75/^1 V - 150/^1^2') + Vx/(11^ w - 5OW2' - 11/^2")- Of the magnitudes required, Z^ is known to us immediately as a function of the X's : * (45) ^' 1728/5(Xj, x^y but the wij, Wj, also readily admit of representation in terms of \ fi. If we introduce, viz., into the defining equations : 2/^ = nij • «,(Xj, Xg) + Tjj . M^(Xj, Xj) • Vy(\, Xj) or : H(\, Xj) 1 /^(^, >-2) • tAh' y • W,{\, X, ) ^(Xj, X,) . r(Xj, X2) the values (40) for the yjs, we get on solution : (46) M. N ,.T(X^,X,) 1 i2/(x„x2)' "' iu.f^(x^,x,y where M^, iV^, denote the following two forms, linear in ^j, /i.^ = /A7N M^i ^i(Xi"_39X,sx^5_ 26x^3x^10) ^ '' M -/x2(26Xiiox/_39Xj6V-X,i3), (48) N^ = ^i(7Xj5x/ + X„') + ^2( - Xj- + 7Xj2x/).* We have now to represent the magnitudes Z^, m^, ?ij, rationally in terms of a, /8, 7, y, on the basis of the formulas (41) — (48) now given.-f- * The magnitudes Z-2, m-2, n^, which are associated with Zi, mi, ni, are omitted for the sake of brevity. + As a verification of the expressions furnished for Mi, A'l, we may observe that the determinant bMi bHj dfj.1 dfjL-2 bNi bNi is simply equal tn II (^l, Xo). 2i8 THE PRINCIPAL EQUATIONS OF § 7. Substitutions of the \, /*'s — Invaeiant Forms. We must now become acquainted with the changes to which \, \, /ij, yiig, are subject when the y's are permuted. These changes are not, however, per se, completely determined. For of the four magnitudes \, //., one is superfluous, even if we take account of the absolute values of the y,'s. We found above that, for the even permutations of the y^'a which we denoted by S and T, -^ undergoes the ikosahedral substitutions so named, while -^ is subjected to substitutions which are derived from these by transforming e into e^- We further remarked that — proceeds from -J- by the cyclic permutation (y^, y^, y,^ y^, and '■2 that a repetition of this operation allows ^ to proceed from — . On the basis of these theorems we shall now define for '^2 \' ^) /"■!. ^2' homogeneous linear substitutions of determinant 1, in such wise that conversely from it, in virtue of (40), the proper permutations of the y^s follow. To this end let us first put, employing homogeneous ikosahedral substitutions of deter- minant 1 : (49) S:X\ = .3Xj, >.'^ -_ ,2?.^ . ^^ = e,ai, ^,' = ^V^ ; [ V5_.X>-(e-.4)X^ + (,2_,3)x^, v/5.XXe2-e3)X, + (.-.4)x^. Vo . ^'i = (£2 - fS) ^^ + (e _ ^4) ^^^ I Vs. ^'2 = (£-£*)//.! -(£2- £3)/^, where the formulae for the /i's again proceed from those for the \'s on replacing e by e^.* Applying these substitutions to (40), there follows in fact necessarily : ,S: 2/; = 2/^+1, T:iji = Vw iJi = 2/2. 2/2'. = 2/. 2/3' = Vv 2/4' = Vi- (50) * Here \/o = ( + e* - e" -^, as must not be overlooked, changes its sign. THE FIFTH DEGREE. 219 Here the permutations of the y's which arise by composition of S and T are, of course, only hemihedrically isomorphous with the corresponding substitutions of the X, /i'b : there are 120 substitutions of the \, /i's, and only 60 permutations of the y's. This circumstance is explained by the fact that, among the sub- stitutions of the X, fi's, the following is found : for which the i/Js remain altogether unaltered as bilinear func- tions of the X, fi'a. We proceed to introduce the following substitution, which we describe shortly as an interchange of X and /i : (51) ;ij' = Xj, 1^2 = A2> \' = y-i' V = ~ Fi- From the formula (40) we then get : yj = y2v, therefore, in fact, the uneven permutation of the yjs previously employed. In agreement likewise with what precedes, we get on repetition of (51) : \ = \, Xj' = - Xj ; ^1' = fij, 11.2 — - /ij, i.e., the homogeneous ikosahedral substitution otherwise denoted by V. Instead of the two-valued or symmetric homogeneous func- tions of the 3/„'s, we shall now fix our attention altogether on such rational and, in particular, integral homogeneous functions (forms) of the X^, X^, as remain unaltered for the substitutions (49), (50), and (51) respectively. If this is only the case for (49) and (50), they are to be called invariants simply, while we will speak of complete invariants if invariance also occurs for (51). It may happen that an invariant merely changes its sign for (51) ; we then call it alternating. If an invariant is neither complete nor alternating, it will, in virtue of (51), be co-ordi- nated with a second. The relation of the two invariants is then mutual, for the repetition of (51) is an ikosahedral substitution, and therefore leads back to the original invariant. Evidently a, j8, and 7 are complete invariants, y an alter- nating one. The forms which we have elsewhere used, /(\, X^, H{X^, \), T(X^, X^), M^, iVj, represent the more general type. 220 THE PRINCIPAL EQUATIONS OF Calling the first three /j, H.^, T^, for the sake of brevity, the forms which are derived by interchange of \ and /x shall be denoted by/^, H^, T^, M^, N^. § 8. General Remarks on the Calculations which we HAVE TO Perform. The statement of the question given in § 6 requires that certain rational invariants shall be expressed rationally in terms of a, yS, 7, V- To this end we may first ask : what integral invariant functions (forms) are integral functions of a, /8, 7, V ' Evidently all those, and only those, which are integral functions of the yja. But these are all such forms as have the same degree in \j, \2) (I'nd /x.^, /x,) respectively. For, on the one hand, every integral function of the yja certainly gives an integral function of the same degree in the X's and the /a's, and, on the other hand, every form of the \, /x's which is of the same degree in the X's and /ix's can be written in the form of an integral func- tion of the terms Xj/Xj, Xj/x^, Xj/Xj, Xj/iXg, and these terms are, disregarding numerical factors, equal to ^j, p^, jOg, ^'4, i.e., inte- gral functions of the yjs,.* On the basis of this theorem, our method will now be to so dispose a given rational invariant, which we are to represent as a rational function of a, yS, 7, V) ^7 t^*® application of appro- priate factors in the numerator and denominator, that the numerator and denominator, taken by themselves, are invariant forms of the same degree in the X, /x's, and then computing numerator and denominator individually as integral functions of a, /3, 7, V- Now, as regards the evaluation of such integral functions, we remark that every invariant form of the same degree in Xj, Xg, and /i]^, /Xg, admits of icing split up into a complete and an alternating invariant. In fact, let F.^ be the proposed form, F^ the co-ordinated form which arises from it by interchange of X and /x. We then simply put : (52) F,=^Eu\^^. Cf. the analogous remark in the last paragraph of the preceding chapter. THE FIFTH DEGREE. 221 Here — ' ^— ^, as a complete invariant, is an integral function of w ~~ w a, 0, 7, alone, while —^ — -, as an alternating invariant, breaks np into the product of V and an integral function of a, /3, 7. The few rules thus established allow us to grapple with the computation of the magnitudes wij, ?ij, ^j, by direct means. § 9. Fresh Calculation of the Magnitude In our new notation : M. (53) m, = ^^^. We will now first multiply numerator and denominator by such an invariant form that there results on both sides the same degree in the X, /a's. It is evidently simplest (though not necessary everywhere) to choose /j as such a factor. We thus write : t^") '■•''-^& In this formula the denominator is in itself a complete invariant ; but we subject the denominator to the splitting-up process just described. We thus obtain : (!50; mi- 24/1/2 the computation of 17^ is therefore reduced to replacing the two complete invariants : 71/1/2 + jl/g/'i aiid /1/2, as well as the alternating invariant : by appropriate integral functions of a, /3, 7, and of a, /3, 7, V, respectively. We solve the problem which now lies before us by taking into consideration, on the one hand, the degree of the forms in the \, fi's which have to be compared, and, on the other hand, returning to the explicit values of our forms in the X, fi's (as we gave them in § 6). The invariants just mentioned (MJ\-, + M.-,/^), 222 THE PRINCIPAL EQUATIONS OF &c., are respectively of degree 13, 12, and 13 in the X, /x's. On the other hand, a, /3, 7, y, exhibit, with respect to the same variables, the degrees 3, 4, 5, 10. Hence we conclude, in the first place, that {M^ifi-^M^f-^ must be a linear combination of the terms a^^, ay^, /S^y, and then, further, that f^f^ is equal to just such a combination of a*, /3^, a^y ; finally, that (M-^f^ — M^f^) coincides with aV, save as to a numerical factor. In order to compute the numerical coeflBcients still undetermined, it is suffi- cient to pay regard to a few terms only in the explicit values of the individual forms, say, then, to the leading terms which present themselves when we arrange the forms according to descending powers of \j and ascending powers of X^. I com- municate here, for the sake of completeness, the leading terms of the forms which we have to consider, each to the extent to which we shall actually want it. We find by § 6 : = W WW + 1 IM1V2' - i^iW') + ^i^'^2 ( - 26;x,iV2' + 39miV/ /i/2 = ^i"^2 W'f^l + 1 WW - i^W) + • ■ ■> ^l/2-^2/l = VV/V2+---, likewise : ay2 = x,l3 (2^^7^^6 + S^XjV') + \'W (0) + • •> ^7 = >-i^^ Wl^2 + SmiVs" + ^iVa") + ^i'^^ (0) + . ., a* = VViV + 4V'ViV+ ••■, ^3= - WlW + 3V^VlV2'+ • • •, oy= _ Xjl3^jl2|,^ + . . . From these values we now infer immediately : M,f^ + M,f, = 11^3^ + 2^2y _ „.^2^ (56) ] /i./2 = »*-^'+«y8y, ^1/2 - ^lA = - "V. and therefore finally : ^^'^ "'»" 24(al-/i3 + „^y) • which is exactly the value communicated in formula (38). THE FIFTH DEGREE. 223 In the same manner we could now, of course, compute n^ and ^j also : the calculations in question would only be some- what more elaborate, because in them we are concerned with constructions of a higher degree in the X, /t's. We shall be able to decompose these calculations, as we always can in similar cases, by proper principles of reduction into a greater number of smaller steps (compare Gordan's work). We do not enter further on this, because we have already, in § 5, obtained simple formulae for n^ and Z^, and the principle of Gordan's method of computation will be suflBciently known by the example of mj. § 10. Geometrical Interpeetation of Gordan's Theory. In the preceding paragraphs Gordan's theory has been ex- pounded from a purely algebraical point of view : we shall bring it closer to our other considerations if we reflect briefly on its geometrical significance. We have here to interpret as co-ordi- nates on the principal surface the ratios \ : \^ and /u.^ : /Xj, as we regarded them in the last paragraph of the preceding chapter. An equation : then defines a curve lying on the principal surface, whose inter- sections with the generators of the first and second kinds are determined as regards number by the degree of i^ in /i and X respectively. If F is an invariant, the curve in question is transformed into itself for the sixty even collineations, and is therefore half regular so far as it is irreducible. The curve becomes regular, on the same condition, if the invariant F is complete or alternating. If we interpret in this sense the invariants occurring in the preceding paragraphs, we are merely led to curves whose sig- nificance is either immediately manifest or is a priori known. The curves a = 0, /3 = 0, 7=0, have come before our notice above as curves of intersection of the principal surface with the diagonal surface.* • Employing for a the representation given in (41), we can now easily prove the assertion previously made, that the curve o = 0, i.e., Bring'a curve, possesses no true doable points, is therefore irreducible, and belongs to the deficiency p = 4. 224 THE PRINCIPAL EQUATIONS OF y = gives a curve which evidently splits up into ten plane portions ; /j = 0, iT^ = 0, T.^ = 0, represent certain aggregates of twelve, twenty, or thirty generators respectively of the first kind. But what do i>/i = 0, iVi = 0, denote? It follows imme- diately from the form of l/j, N.^, that we have to do with curves of the fourteenth and eighth order respectively, which cut the individual generators of the first kind only once. These are the same curves which we have before represented by formulae of the following kind : (58) py. = t,{\,h)-'^''(\'h) In fact, we shall be led back to these formulae if we determine from M, = 0, or N, = 0, the '^ as a rational function of -^, and insert the valne found in the formulae (40) : In the same sense the equation : (59) m . T {\, X^) . iV^i + 12n ./2 (X„ X^) . J/j = represents the whole family of those curves of the 38th order which we considered in § 4 of the present chapter (see formula (27)). We now turn in particular to the computation of m^ given in the preceding paragraph. Originally we had by (53) : m, is therefore a function on the principal surface, which vanishes along the curve of the 14th order, il/^ = 0, and becomes infinite for the twelve generators of the first kind/j = 0. Writing now, as was done in (54), '"'"12/,/,' we have evidently raised the two curves ifj = 0, /j = 0, by addi- tion of the curve /2 = 0, i.e., an aggregate of twelve generators of the second kind, to the complete intersection of the principal surface with the accessory surface ; the intersecting surfaces can THE FIFTH DEGREE. 225 then, in particular, be so chosen that they themselves are trans- formed into themselves for the 60 even collineations, and are therefore represented by equating to zero integral functions of °-i /3, 7, V- Hence the structure of formula (57), and also the measure of its arbitrariness might be made manifest. I leave it to the reader to interpret in a similar manner the significance of formulae (36), (37), for Z^ and Wj. § 11. Algebraical Aspects (after Gordan). We have so far expounded the Gordan theory as it originated, viz., as a direct method for computing the magnitudes occurring in the solution of the principal equations of the fifth degree. Herr Gordan has, however, in his exhaustive memoir published in the 13th volume of the Annalen, chosen a much higher stand- poLut ; he has proposed to himself the problem : to construct the full system of invariant forms F(k.^, Xj ; /^i, ^2)1 ^^^ ^^ many relations as possible between these forms. He thus finds 36 systems of forms, of which those which are different from a, /9, 7, V) ^re connected by permutation of X, and fi. We cannot go more fully into these results, but must consider the method which Herr Gordan has employed for their deduction. Let us recall how we previously deduced S{\, Xg), T{\, X^), from f{\, X2)) "^y nieans of processes of differentiation appertaining to the invariant theory. In just the same way Herr Gordan obtains his forms, putting at the head of them : as a " double-binary ground-form with two series of indepen- dent variables." Let us first explain, in respect to this, how f{\, \) [the ground-form of the ikosahedron] is now to be defined. Consider Xj, X2, as constant in a, i.e., a as a binary form of the third order of /ij, /i„ only. Then, I assert, / is the discriminant of this form of the third order, disregarding a numerical factor. We confirm this by direct calculation. We first construct, in accordance with the usual rules, the Hessian form of a, and find, save as to a factor, the following invariant, quadratic in the ^'s : P 226 THE PRINCIPAL EQUATIONS OF (60) ^ = MiM - \' - 3XiV) + 10/xi/x2\sx„^ + /./ (3aj5x, - V), which we shall again use later on. We further compute the determinant of t, and come back, in fact, disregarding a numeri- cal coefficient, to : /= XjiiXj + iix,6>./ - Xj yi. Let us further explain how Herr Gordan obtains the formulae of inversion, which we were able to establish in § 4 by applying those data which we obtained above, I. 4, § 12 (somewhat in- cidentally), by the formation of the principal resolvent of the ikosahedral equation. In Herr Gordan's method those in- variants which are linear in yu.j, fj,^, form the starting-point. He shows that four diflferent invariants of this kind exist, among which those two which are of lowest degree in the \'s are exactly identical* with our iVj, My Now by formula (40) the yjs are themselves linear forms in /Xj, fi^ : Hence we can write, from the outset : (61) ayy=h,.M^ + c,.Ny where the coefiBcients a,, &^, c^, are to be taken from the identity: bfj,^ d/J-^ d,a^ =0. by, bM^ bN^ biMr, bfi^ b/j,^ Here a, as the functional determinant of Jfj and N^ is itself an invariant; we have seen above that it is identical with H (Xj, Xj)- On the other hand, &„, c„ are necessarily five-valued, like the yjs themselves. Computing them as functional deter- minants of y, and iV^, y, and M^ respectively, we then get the same magnitudes as we before denoted by JF„(Xj, \^ and i^X,, ^2) • ^^■'{'^1) ^2)- 1° f^*^*' formula (61), written in our earlier notation, must run as follows : * One of these four invariants, if we multiply it by H^, is contained in the general form m. Ti. Ni + I2n.f^ .Mi, the vanishing of which represents those curves of the 3fcth order which we have previously considered. Among these curves, in addition to Mi = 0, iVj = 0, a third presents itself whose order reduces to a lower number, viz., to 18. THE FIFTH DEGREE. 227 (62) H{\, \) . y, = W,{\, X2) . ifj + U(X^, A^) . W,{\, >^).N,; compare, say, formula (46) supra. We can say that Gordan's development of this formula, as we have explained it, is just the reverse of ours. The further course of the calculation is then the same in both. In order to express the i/Js by means of the X,'s and the other given magnitudes, we introduce in (62), instead of 31^, iVj, the expressions : i.e., quotients which are both of the first dimension in X^, \„, and ^j, fi^, and then compute these as rational functions of "j A 7i V) i^ *^6 same way as was done in § 9 for m.^ in particular. We pause for another moment over Gordan's derivation of formula (62). We can evidently put it into words as follows. Since the i/^'s are bilinear forms in \, \,, and /i^, [i^, their determination requires (if we have assumed Xg arbitrarily — as is allowed — and then found \ : Xg from the corresponding ikosahedral equation) only the knowledge of /ij, /tg, in addition. JVe now obtain these by annexing the two invariants, linear in ft^, /J.2, to wit, M^, A\, and compute them as rational functions of X^, Xg, and of a, ^, y, '^. In fact, we have thus two linear equations for /ij, fj,^ ; if we solve these for /ij^, fj,^, and insert the values which arise in the formula for 2/„, we have the result which we sought, the same which is presented in an abbreviated form by (62). Or we can also put it thus. If we put 3/^ = 0, we determine in the binary manifoldness fi^: ii^a first element contragredient to the elements Xj : Xj, or — to express it more generally — a covariant element. We obtain a second element of the same kind if we take N.^ = Q. Our problem is to find that element in the manifoldness ftj : fi^ which is represented by 2/„ = 0. We solve this problem in (62) by composing y^ with the two covariant elements M^ and A\ by the help of appro- priate coefficients, thus proceeding according to the same fun- damental theorems of the "typical representation" which we employed above in describing the Tschirnhausian transformation. The mode of conception thus denoted will often come into play later on in a generalised form. 2 28 THE PRINCIPAL EQUATIONS OF § 12. The Normal Equation of the rja. In our general survey of the different paths struck out for solving the equation of the fifth degree, we have above (II. 1, § 1) separated the method of resolvent construction from that of the Tschirnhausian transformation, remarking, however, that we can always replace the one method by the other. When we solved the principal equations of the fifth degree directly by the help of the ikosahedral equation, we followed the method of resolvent construction. If we are to expound the method of the Tschirnhausian transformation in place of it, we shall have to start from one of the resolvents of the fifth degree, which we have established in I. 4, for the ikosahedral equation, as a normal equation. The resolvent of the rjs which we constructed in § 9, loc. cit., and to which we then imparted the form : (63) Z:Z-1 :l = (r-3)3(r2-llr+64) :r(r2-10r + 45)2 : - 1728, seems to be in this respect most adapted to our purpose. In fact, we have already (supra, § 13, loc. cit.) represented the m„, v„ rationally in terms of r^ : 12 12 if we insert these formulae in our present one : l/y = ?nj . M„ + Tlj . lly . Vy we obtain immediately the representation of y^ by means of the roots of the normal equation (63) : _ 12(r,-3K+144wt ^^^^ ^''~(r,-3)(r„2-10r„ + 45)- The only further question that arises is how we are to compute JAX„\) as a rational function of the yy's. We will here strike out a path similar to that just taken (| 9) in the computation of wij. For brevity, let : THE FIFTH DEGREE. 229 then we write in turn : ''"/r/2 _ L^" ' 1 ' /a "*" ^y ' 2 ' / l J "^ L ^" I I ' J'2 ~ ^y I 2 ' /i J Here /^'/g is, as we know, equal to (a*— ^8^ + 0/67). We can now compute, in a perfectly analogous manner, the two por- tions of the numerator (by returning to the explicit values in \, Xg and ^j, /ij). Let us for a moment suppose the y's to be introduced in place of the \, fiB, then these components are such integral functions of the y'a as remain unaltered when we permute those four y's, which are different from our fixed y„ in an arbitrary manner and in an even manner respectively. Now the sums of the powers of these four y's are integral functions of y„ a, /8, 7, but their difference-product is equal to 5v:(2// + 2ay„-t-;8), where (y* + 2ay+^) denotes the differ- ential coeflBcient of the left side of our principal equation divided by 5. Hence p^^ l■/2 + <.-^ 2"/i] '^^^^ ^® ^° integral function oi y„ a, /S, 7, but [tj^, \- f2~^^'i 2'/2] ^'"^ ^P^^* "P ^^*° the product of such an integral function and the magnitude —7 — ~ -r.- It is not necessary for me to go into the details y, + lo-y^ + p of the calculation ; I will therefore only communicate the result.* We find : (65) 2(a*-^3 + „j8y),.^ = [(ay + 2/32) y^ + ,„3 _ ^y) y% _ ^^i^ . y^ + {^a^y + Ucc^-) J,„ + (lla^ . 9a;3y)] - [(a,„3 , ^y. ^ „2) . ^^ V__J. Summing up, we have the following result : We have in (65) the Tschirnhaudan transformation which transforms the given principal equation into the normal equation (63) ; if we have then determined the roots r^ of the latter, (64) gives tis the explicit values of the yjs lohich we sought. * See Math. Ann., t. xii. p. 556. 230 THE PRINCIPAL EQUATIONS OF § 13. Being's Transformation. I have communicated in detail the formulsB of the preceding paragraph the more willingly because from them, as I shall now show, all formulas can be derived which are required in the execution of the Bring transformation* Let y^, y^, y.-,, y^, y^ and y^, y^, y^\ y^, y^ be the co-ordinates of two points on the principal surface which belong to the same generator of the first kind. Then we obtain for the corresponding principal equations the Z and »v's,f while we distinguish the other magnitudes which present themselves therein for consideration by addition of an accent, and will therefore put a\ /S', 7', Sj', wij', «i' in the second equation opposite a, /3, 7, y, Wj, Wj in the first. I say now that a double application of the formulae (64), (65) is snfli- cient in order to transform one of the principal equations into the other, and the roots of the one into those of the second. We will, for brevity, denote by (64'), (G5') the equations (64), (65) when they are written with accentuated letters. Then the whole process which is here necessary evidently consists in expressing first, by means of (65) the r„'s in terms of the yjs, and then, by means of (64'), the y"s in terms of the ?'„'s (which is the transformation we sought) ; and then, conversely, com- puting by means of (65') the r^'s as functions of the yjs, and so finding from them the yJs by means of (64). Tlie Bring theory is furnished iy a special case of the getieral method thus given. The generator of the first kind, viz., which carries the point y, meets the curve a = in three points : we obtain the Bring transformation if we choose one of these points as y'. This means, analytically, that we are so to determine wij', 11-^ that, in the principal equation for y', the terra involving 2/^^ disappears. A glance at the general principal resolvent (I. 4, § 12) gives us at once the cubic equations which m-^, n-[, must satisfy in consequence ; in other words, the cubic auxiliary equation which the Bring theory required ; it is as follows : (66) 8»i3 + I2m2« + "^f = 0. * See the analogous formulae iu Gordan's paper in Ed. xiii. of the Math. Ann., p. 400, &c. t Or more correctly Z\ and r„ I's, as we might have written in the preceding paragraph. THE FIFTH DEGREE. 231 It depends, as is clear a priori, not on the individual point y, but only on the generator of the first kind on which this point is situated, and on the sixty generators which arise from the one mentioned in virtue of the even coUineations. We have nothing further to add concerning the Bring theory ; the most we can do is to call attention to the fact that (65') now be- comes very simple, inasmuch as a' = 0.* It will also be useful to give prominence to the fact that, in the trinomial equation which we obtain by carrying out the Bring transformation, we always know, a priori, the square root of the discriminant. § 14. The Normal Equation of Hermite. Now that we have brought the Bring theory so simply into connection with our developments, we will seek to do the same with the normal form on which Hermite bases the solution by elliptic functions. As we saw above (II. 1, § 4), this runs as follows : (67) r5-2*. 53 . M* (1 - mS)2 . F-2«VP.M='(l-M«)Ml+«^) = 0, where ?t^ = k^. We shall inquire if this equation is contained as a special case in the general principal resolvent of the ikosa- hedral equation, when we put Z, the right side of the ikosahedral equation, equal to : ^^^ A 27 ■ x4 (1 - x2) •1 y as we did above (I. 5, § 7) in dealing with the solution of the ikosahedral equation by elliptic modular functions; we shall inquire why Hermite in his investigations was led, at the outset, to the Bring form, while every principal equation of the fifth degree can be solved by the help of elliptic functions (through the intervention of the ikosahedral equation), and the Bring form is by no means the simplest among the infinitely many * In a manner similar to that in which tlie Bring transformation is effected by means of (66), the problem is solved by the help of an equation of the fourth decree ; from the given principal equation to establish another for which /9' = 0. To the feasibility of this problem it seems that Jerrard first called attention [Mathematical Researches, 1834]. 232 THE PRINCIPAL EQUATIONS OF principal equations with one parameter wMch present them- selves. In order to answer these questions, let us insert for Z in (66) the function of k^ given in (68). The result is that the ciibic equation (66) hecomes reducible. In fact, it is satisfied, as we can verily immediately, if we choose : 7«:ra = 3x2:2(2-5;t2 + 2K*). I will accordingly put : (69) m = 3x2(l + x2), M=2(l+>c2)(2-5x2 + 2x4). The coefficients of the principal resolvent given in I. 4, § 12, are then considerably condensed, so that we obtain the equa- tion: (70) 2/5 _ 24 . 38 _ 5 . xio (1 - (c2)2 . 2/ - 26 . 310 . jci2 (1 _ x2)2 (1 + x^) = 0. Here we need only further substitute for y : (71) y=—^-y, 9x^ in order to find precisely the Hermite equation. Our first question is therefwe to he answered in the affirmative. At the same time we discern the answer to the second question in the circumstance that Hermite operated, not with the rational in- variants g^, g^, hut with k^ throughout. If we now compute for the Hermitian equation, or, what comes to the same thing, for (70), the corresponding ^j, we naturally come back, a proper choice of the sign of V being made, to ?!-.* But a very simple value arises for Z^ also ; we find, on reversing the sign of V in the expression for .Zj : ^'•^^ ^2- 108x2 (l_x2^4-T This, as is shown in the theory of elliptic functions, is one of the a 3 three values which arise from ^ by a quadratic transformation of * We have here to take (for (70)) : V=2" .3^.k" (1 -k')* (1 - 6ic' + K*). + Cf. Oordan, loc. ait., or my communication, already mentioned, in the Rendi- conti of the Istituto Lombardo of April 26, 1877. THE FIFTH DEGREE. 233 the elliptic integral. We cannot, unfortunately, follow up further in this place the interesting connection of the Bring curve with the quadratic transformation of elliptic functions which here presents itself.* Breaking off for the present these developments, we here content ourselves with the fact that the Briog and Hermitian formulae fit in with ours. In the fifth chapter we shall return to our present results from a general point of view, and seek to decide what theoretical value they possess. * Cf. my memoir : " TJeber die Transformation der elliptischen Functionen und die Aufldsung der Gleichungen fUnften Grades," in Bd. xiv. of the Math. Annalen (1878), especially p. 166, &c., of the same. ( 234 ) CHAPTEE IV. THE PROBLEM OF THE A's AND THE J AGO BI AN EQUATIONS OF THE SIXTH DEGREE. § 1. The Object of the Following Developments. In the preceding chapter we have considered two series of binary variables, \, \-: ^^^ l^v l^v which were simultaneously subjected to homogeneous ikosahedral substitutions, and besides to a process which we called the interchange of \, ft. We have further had under investigation certain bilinear forms of the X, /a's which we called i/^. The yjs undergo on their part, for the transformations of the X, /t's in question, linear substitutions of the simplest possible kind, to wit, mere permutations, and indeed permutations of the whole set ; if we are therefore to establish a corresponding form-problem of the y's, this finds its complete expression in the equation of the fifth degree which the yjs satisfy, i.e., in the principal equation. We can in this sense assert that we have been concerned in the preceding chapter with a form- problem which arises from the consideration of the simultaneous substitutions of the X, ju.'s. Now in the following pages a statement of the question of a quite similar kind (which moreover possesses essentially a still more simple character) is to be dealt with. The simultaneous ikosahedral substitutions of the X, /i's were, as we called it, contragredient ; ive will now take into consideration two series of binary variables : ivliich are in each case simultaneously subjected to tJie same ikosa- hedral substitutions, and so can he described as cogredient. In the case of these, again, we construct certain bilinear forms, viz., the symmetric functions : THE PROBLEM OF THE A's, ETC. 235 (1) Ao = - ^(XjX/ + \^\-), Ai = w, A2 = - x,x;, i.e., the coeflScients of that quadratic form : (2) AiZj2 + 2AoV2-A2z/, which arises on multiplying out the factors : If we subject the X, \"s to the 120 homogeneous ikosahedral substitutions, or interchange them with one another, these A's undergo on the whole sixty ternary linear substitutions, for the individual A's remain altogether unaltered, not only for the interchange of the X, X"s, but also when we simultaneously reverse the signs of Xj, X^, X^', X^'.* We shall deal with the ternai-y form-proUem which is involved in the consideration of the stibstiiutions thus defined. We have already stated that this form-problem of the A's is essentially more simple than that of the y'a. In fact, we shall be able to return all through with our reflexions and calculations to the ordinary ikosahedral problem, from which the results we seek then offer themselves in virtue of a definite principle of transference well known in modern algebra, so that indeed the accomplishment of our problem appears almost as an exercise in the application of certain fundamental theorems appertaining to the theory of invariants.-f- According to the same scheme, we should also be able to deal with the case of 3, 4 . . . series of binary variables which are subjected to the ikosahedral substitutions or any other group of binary substitutions in a cogredient manner. If among these infinitely many, so to say, associated form-problems we select the one just described, it is because we employ it in the further consideration of equations of the fifth degree. We shall soon learn that the general Jaco- bian equations of the sixth degree, hy which Kroneckers theory of eqimtions of the fifth degree is supported, are resolvents of our problem of the A's. By substituting for it altogether the pro- * The substitutions of the A's are hence holohedrically isomorphous with the sixty ordinary non-homogeneous ikosahedral substitutions. + The principle of transference in question is essentially the same to which Hesse has devoted a memoir in Bd. Ixvi. of Crelle's Journal (1866). 236 THE PROBLEM OF THE A's AND blem of the A's, we shall sacceed in the simplest way in under- standing from our standpoint the various results which have been discovered in other quarters for the Jacobian equations of the sixth degree, and thus in attaining for the general treat- ment of equations of the fifth degree a uniform basis, which is nothing else than a rational theory of the ikosahedron.* The arrangement of the subject-matter for the following developments is already given by what we have said. The first thing is to establish the problem of the A's in an explicit form, where we shall again make free use of geometrical interpreter tion. On then studying the corresponding resolvents, we are enabled to pass over to the Jacobian equations of the sixth degree, and to the researches of Brioschi and Kronecker relative thereto. I finally apply myself to the solution of our problem, and show that it can be accomplished with the help of an ikosahedral equation and an additional square root, in strict analogy with the Gordan theory expounded in the preceding chapter.t § 2. The Substitutions of the A's — Invariant Forms. In order now to determine explicitly the substitutions of our A's, let us recur to the generating ikosahedral substitutions S, T, and U respectively. We had for the Xj, Xj's : (3) r s T U * Like the Jacobian equations of the Eixth degree, the general equations of the (« + !)"' degree which we described above (II. 1, § 13) can be replaced by parallel form -problems which are related to the ^ — ^ — - variables Ao, Aj, . . . An-i. I 2 2 have accomplished this for » = 7 in Bd. xv. of the Math. Ann. (1879); see in particular pp. 268-275. t The principal reflexions to be employed in the following expositions were laid before the Erlanger Societat by me on November 18, 1876 [" Weitere Unter- suchungen iiber das Ikosader, I."] ; cf. further the second part of my memoir, which appeared under the same title in Bd. xvii. of the Annalen (1877). The developments § 8-13 were then added for the first time. THE yACOBIAN EQUATION. 237 Writing* down the same formulas for the \, \, we obtain from (1) for our A's the following substitutions : S: Ao^=A„, A; = £4A, A2' = .A,; |'V5_.Ao'= Ao + Aj + Aa, (4) j T:/ V5_.Ai' = 2Ao + (€2 + .2)Ai + (. + c*)A2, ( V5 . A2'= 2Ao + (€ + .4) Aj + (£2 + e3) A^ ; U: Ao'=-A„, A/=-A2, A2'= -Aj, which all, like (3), have the determinant + 1. From them are composed the 60 linear substitutions of the A's which exist according to the old scheme (I. 1, § 12): (5) ^^ S>^T&', S>^U, Si^TS-Ui^, >= 0,1, 2,3, i). Now, as regards the invariant forms, i.e., those integral homo- geneous functions of the A's which remain unaltered for the substitutions (5), the determinant of (2) : (6) ^=Ao2 + AA, at all events, belongs to them. In fact, this becomes, on intro- ducing the X, \"s, equal to (Xj, X^-X^, \{f, and therefore re- mains altogether invariant if we subject the \, X"s simultaneously to any homogeneous substitutions of determinant 1. Besides A, the full system, of the forms which we seek ivill only contain, as I assert, three more forms, of the 6th, 10th, and 15th degrees re- spectively. Namely, if ^ = 0, then \' = M\, X^' = M\, under- standing by Man arbitrary number ; therefore by (1) : (7) A,= -M>.,\„ Ai = l/V, A„=-mi2 The required forms are accordingly transformed into multiples of forms of \, Xjj whose degree in the X's is double as great as the original degree in the A's, and which, moreover, have the property of being transformed into themselves by the homo- geneous ikosahedral substitutions of Xj, >2- ^^^ ^^ow the system of all ikosahedral forms is constructed by the form of the 12th order f{\, Xg), the form of the 20th order S(X^, Xj), and the form of the 30th T(\, Xj). Hence follows our assertion by * I hope no misunderstanding will arise from the fact that the letters \i, X^, just employed in formula (3) on the left-hand side have been used before, of course with quite another meaning, 238 THE PROBLEM OF THE As AND reciprocation. We might even say that, corresponding to the identity : (8) r2= 1728/5 -J3-3, a single identical relation will subsist between the new forms which is transformed into (8) as soon as we put ^ = 0. I will denote the three required forms by £, C, B. In prov- ing their existence by reverting to the ikosahedral forms/, H, T, we have already made use of the algebraical principle of transference which we proposed above. This we shall do in a higher measure by now actually establishing B, 0, B, if only in a provisional form. We are here dealing with a process of polarisation adapted to the purpose. If (\, \) i^ ^^7 fo^'^i which remains unaltered for the homogeneous ikosahedral sub- stitutions of Xj, X2, and if \', X^^ ^^^ cogredient with \, \j then all the polars : will be invariant for the simultaneous substitutions of the \, X"s. Let us now construct in particular for /j (Xj, Xg), II{\, Xj), T (Xj, Xg), respectively the siKth, tenth, and fifteenth polars. We thus obtain invariant forms which are symmetrical in the X, X"s, and therefore represent integral functions of A^, A^, A^. On writing them down as such, we have found the required forms B, C, B. In fact, these forms are now necessarily invariant for the substitutions (4) or (5) ; they have, moreover, the degrees 6, 10, 15, in the A's, and are transformed into multiples of/(X^, Xg), If (X[, Xg), I'(Xj, X2), when the formulae (7) are applied. I will communicate here at once the result of the calculation. After separating particular numerical factors, we find in the manner explained : B' = I6A06 - ISOAo^AjAj + gOAo^Ai^A^s + 2\^^{^^^ + A/) - SAi^A/ C" = - 512Ao'o + 11520A(,«AiA2 - AQ'i1Q\^^^^^K^^ + SSeOOA/AjSAj^ - eSOOAo^Ai^Aj* - 187 (AjW + Aji") + 126Ai5A/ (9) I +Ao(Ai6 + A/) (22176Ao^ - 18480Ao2AiA2 + IQSOA^^A/), Z) = [Ai6 - AjS] { - IO24A0I'' + 3840Ao«AiA2 - ZMQ^^^^^^^^ + 1200Ao*Ai3A/ - lOOAo^Aj^A^* + Ai^" + K}" + 2k^^t<^^ + Ao{A,5 + AjS) (352Ao* - leOAo^AjA^ + lOAi^A^^)}. THE yACOBIAN EQUATION. 239 I have here denoted the two first forms no longer by £ and C, but by B" and C, because I will hereafter modify these further by addition of factors which contain A as factor. Only when this is done shall I establish the relation which makes B^ equal to an integral function of A, B, C. If we apply the substitution (7) to the preceding forms, where for simplicity we will put M= 1, we have, in agreement with what was said before : iB'= 21./(x„x,), (10) {C' = \&7.H{\,\), § 3. Geometrical Interpretation — Eegulation of the Invariant Expressions. In order to facilitate our mode of expression and the growth of our ideas in the domain of the function-theory, we now introduce our geometrical interpretation. Eetaining through- out the analogy with the developments of the preceding chapter, let us regard A^ : Aj : \ as the projective co-ordinates of a point in the plane, the substitutions of the A's as so many coUinea- tions.-)- The individual invariant form of the A's then repre- * The method of calculation contained in the text is described in the text- books on the theory of invariants, at the suggestion of Gordan, as transvcction of the quadratic forms (2), and indeed (disregarding numerical factors) B' is the 6th, C the 10th, D' the 15th, transvectant of the corresponding power of (5) over/, If, and T respectively. I have not applied this mode of expression and the corre- sponding symbolical relation in the text, because I wished not to presuppose in this respect any specific preliminary knowledge on the part of the reader. + We can of course regard every form-problem in a corresponding manner. If we proceeded differently in the foregoing part, and interpreted the binary form- problem by means of points of the {x + iy) sphere, it was because we wished to have intuitively before our eyes not only the real, but also the complex values of the variables in the elementary sense. I annex hereto a somewhat different interpretation of the problem of the A's. Put A(i=z, ^i = x + iy, Pi'i = x-iy, and regard x, y, z, as the rectangular co-ordi- nates of a point in space. Observing that the sixty substitutions of the A's have the determinant unity, and A is now = xr + y- + z^, we recognise that the said substitutions now correspond to rotations round the origin of co-ordinates. These are the rotations for which a determinate ikosahedron is brought into coincidence with itself. The six fundamental points to be immediately introduced in the text give on this interpretation those six diameters which connect two opposite summits of the ikosahedron. On the other hand, the equation D = (of which we 240 THE PROBLEM OF THE A's AND sents, when equated to zero, a plane curve which is transformed into itself for the said collineations. In this respect we have first the conic ^ = 0, which we will call the fundamental conic. If we write in accordance with formulae (7) (again taking we have expressed the variable point of this conic by means of a parameter r^. Hence we shall be able to denote the two parameters -^ ^, which appear in formulae (1) by means of two points of the fundamental conic. These are the two points in which the two tangents from the point A to the fundamental tonic touch the latter. In fact, the polar of the point A with respect to ^ = has the equation : 2AoAo' + A2A/ + AA' = 0> and this equation is satisfied if we substitute for the A's the expressions (1), and for the A"s the expressions (7) or the corresponding ones in which X' is written instead of X. The points of the fundamental conic are naturally so grouped that aggregates of twelve, twenty, thirty of them are self- conjugate, represented respectively by : /(Xj, X^) = 0, fi-(Xj, >^) = 0, T{\, \) = 0; these are at the same time the points of intersection of ^ = with the curves 5=0, C=0, D = 0. We will now connect by a straight line those pairs amongst these points which remain fixed for the same collineations. Then we obtain, corresponding to the forms /, H, T, six, ten, and fifteen straight lines respec- tively. Constructing, then, for this line its pole with respect to the fundamental conic, we obtain self-conjugate groups of six, ten, and fifteen points in the plane. Let us now consider the form of the equation : 4 = Ao2 + AiA2 = 0. Bhall presently show that it splits up into linear factors) gives the fifteen planes of symmetry of the configuration. | We can combine this new interpretation with that of the \, X' 's on a sphere, but I do not enter upon this, since it would lead us too far. THE yACOBIAN EQUATION. 241 Clearly, the two angles of the system of co-ordinates : Aq = 0, Aj = 0, and Aq = 0, Aj = 0, which appertain to A = 0, are corresponding vanishing-points for/, for both remain unaltered for the collineation S [see supra, formula (41)]. Therefore Ag = is one of the six straight lines which belong to /; Aj = 0, A2=0, the corresponding pole. In agreement herewith A^^ assumes, for our sixty substitutions, only the following twelve values, each pair agreeing as to sign : (11) ±Ao. ±(Ao + e-'Ai + €*''A2); and in accordance with the same formula only the following five points are grouped with the point Aj = 0, A2 = : (12) Ao:Al:A2=l:2£*'■:2«^ I will describe the six points thus distinguished as fundamental points of the plane. If we connect a first fundamental point with the five others, we obtain the five straight lines : {"Aj - (^"A^ = 0. Evidently the left sides of this equation are all contained as factors in the value of D just communicated. The curve J):=0 must, however, be uniformly related to all the fundamental points. Hence the curve D = splits up into the fifteen connecting lines of the six fundamental points. The following algebraical decomposition corresponds to it : (13) D = Jlie'A, - ^-A,) ■ II((1 + ^^5) Ao + ."A, + .^A,) V V .IX((l-s/5)Ao + e*>'Ai + .-'A,), (v = 0, i, 2,3, 4), as we easily verify. We could establish a number of interesting theorems about the fifteen straight lines here presenting them- selves ; they are the fifteen lines which belong to the point-pairs of T ; they pass in threes through the ten points which we co-ordinated * to the point-pairs of H, &c. I do not enter * Clebsch has incidentally dealt with the figure described in the text in the course of considerations allied, though ag.iin formulated quite differently, and has thus announced the last-mentioned property : the six fundamenlal points form a Q 242 THE PROBLEM OF THE AV AND further here on these theorems, because we do not make further use of them ; moreover, they are easily recognised as trans- ferences of properties of the grouping of points which occur in connection with the ikosahedron. As regards the curves ^ = 0, C' = 0, these have no special relation to our six fundamental points. It is this circumstance ivhich we will now make use of in order to replace B' and C iy two other easpressions. We shall introduce instead of B' a linear combination B of B' and A^ in such wise that the curve B=0 contains the fundamental point Ai = 0, A2 = 0, and therefore (as an invariant curve) all the fundamental points. In the same way we shall replace C" by a linear combination C of C, A^B, and A^, which, equated to zero, represents a curve which has at Aj = 0, A2 = 0, and therefore at all the fundamental points a singular point of the highest possible kind. In this manner we find (after casting out particular numerical factors) : / — B' + 16 A^ iB= 21 = 8A„*A,A2-2Ao=Ai2A/+Ai3A/-A,(Ai=+A/), /-. . ^ Ly - C - 512A^ + 1760^2^ (14); (7= ^g^ = 320Ao«Ai2A/ - leOAo^AjSA/ + 20Ao2A,*A2* + GAi^A/ - 4Ao (A/ + A/) (32Ao* - aOAo^AjA^ + SAj^A/) + Aji" + Aji<>. Evidently B=0 has at Aj = 0, A2 = 0, and thus at all the funda- mental points not a merely ordinary point, but a double point, and is therefore (since we can show that it can possess no further double point) of deficiency 4. Similarly C=0 has at each of the fundamental points two cusps, i.e., a 4-tuple point, and is therefore of deficiency p = 0. If we substitute in our new B, C, in accordance with for- mula (7) : Aq = — '-1^2' Aj = Aj > Aj = — \ , we have : (15) B=-f(X„K), C=-^(Xj, >2), which we may compare with (10). The relation which expresses tenfold Brianchon hexagon (Math. Ann., Bd. iv. -. "Ueber die Anwendung der quadratischen Substitution auf die Gleichungen 5. Grades und die geometrische Theorie des ebenen Fiinfseits," 1871). THE JACOBIAN EQUATION. 243 Z)2 as an integral function of A, B, C, will therefore have the following terms not involving A : D^= -172855+ as. Returning to the explicit values (9), (14), and taking account of a sufficient number of terms, we find the complete formula : * (16) Z?2= - 1728B5 +0-3 + 7204 C5'' - mA^C^B + 64^3(5^ - ACf. § 4. The Problem of the A's and its Reduction. The problem of the A's, as we proposed it, is fully determined by the explicit formulae (6), (9), (14), now obtained for A, B, C, D, and the relation (16). We suppose the numerical values of A, B, C, D, given in some manner in agreement with 16 ; our problem requires us to determine the corresponding systems of values of A^, Aj, Aj. Since A, B, C, D, form the full system of the invariant forms, our problem can only possess such solu- tions as proceed from some one thereof by the 60 substitutions (5). In fact, if we determine the number of solutions by Bezout's theorem, we shall be led to the number 60. Namely, from the values of A, B, C, arise at the outset 2 ■ 6 • 10=120 systems of values of the A's, of which, however, since A, B, C, are all even functions of the A's, certain pairs can only differ by a simultaneous change of sign of the A's. Of these 120 systems of values, only half can therefore satisfy the given value of D, since D is of uneven order. All CO systems of solution, as has already been said, proceed from some one thereof by the sub- stitutions (5). We can therefore say, in the sense explained previously (I. 4), that our problem is its own Galois resolvent after the adjunction of e, and therefore possesses a group which is holohedrically isomorphous with the group of the 60 ikosa- hedral rotations. We consider now, in reliance on I. 5, § 4, the parallel system of equations. The ratios of A^ : Aj : A, are evidently determined in sixty ways if in the equations : * Cf. Brioschi in torn. i. of the Annali di Matcmatica (aer. 2, 1867), p. 223. 244 THE PROBLEM OF THE As AND we can regard the values of Y and Z as known ;* the required points A are the complete intersection of the curves of the sixth and tenth order respectively : From the 60 solutions of the equation system we now com- pute those of the corresponding form-problem rationally. Put, namely : (18) J. = ^. Then if A^ : A^ : A^ = a^ : a^ : a^ is one of the systems of solution of the equation problem, we have evidently : (19) Ao = /)«(,, Ai = pai, A^_ = pa^ understanding by p the following expression : _ ^' («o, «i, « ;) X P' D{u„a„a,y^' whereupon the statement is proved. In this respect an essential difference exists between the binary form-problems previously studied and the present ternary ones ; for then we required, as we showed in I. 3, § 2, one addi- tional square root in the supplementary solution of the form- problem. This, of course, corresponds to the circumstance that the group of the homogeneous binary substitutions was only hemihedrically isomorphous with the non-homogeneous ones, while now holohedric isomorphism occurs. On the other hand, the two agree in another point. We could in the former case reduce the form-problem, as we called it, i.e., replace the three magnitudes i^j, F^, F^ (connected by an equation of condition) on which the form-problem depended, by two independent variables, X and Y, which were themselves rational functions of F^, Fo, F^ ; whilst conversely the latter again depended rationally on them. We obtain just the same result in the problem of the A's if we take into consideration the quotients X, Y, Z, which we just introduced in (17), (18). These magni- tudes X, Y, Z, are in themselves defined as rational functions * These magnitudes, 1', Z, are the saiu^ that we have denoted by a, b, in II. ], § 7 [formula (36)]. THE JACOBIAN EQUATION. 245 of ^, B, C, D, but we can conversely also express A, B, C, D, rationally by means of X, Y, Z. In fact, if we divide in (16) both terms by A^^, we have, after an easy rearrangement, in virtue of (17), (18) : (20) A = ~ ^ ^ Z3-1728r5 + 720r3Z-80rZ3 + 64(5y2-2)2' while (21) B=Y-A^, C = Z.A\ D^^^X-A'', which are the desired formulae. It is instructive to bring forward also for comparison here the problem of the yja, which we studied in the preceding chapter as the principal equation of the fifth degree. We then supposed that, in addition to the coefficients a, ff, 7, of the equa- tion, the square root V of the discriminant was given, the square of which is an integral function of a, /3, 7. We then obtained 60 systems of solution jr^, y^, y^, y^, y^, which were again fully determined (rationally) in terms of the correspond- ing values of the ratios 2/0 ■ 2/1 • 2/2 • llz '■ Vi- This depends on the fact that we can construct as before from the given magnitudes By. quotients, e.g., — or 4, which are of the first dimension in tbe y's. We can also reduce the form problem of the y's, only this is not so simply attained as in the other cases. The reduction is actually given by the m, n, Z, of the principal resolvent of the ikosahedron. We have represented, viz., in T. 4, | 12, § 14, a, yS, 7, Vj rationally in terms of vi, n, Z, while conversely we have just now (in II. 3) given exhaustive methods in virtue of which m, n, Z, appear as rational functions of a, /3, 7, y . If -4 = for the form-problem of A's, we can solve it directly by means of the ikosahedral equation : namely, if we have determined \ : X, from it, we find by formula (7) : Ao : Ai : A2 = - \K ■.\-:- \-, and hence, as we saw above [formula (19)], the values of Aj, Aj, A^ themselves. 246 THE PROBLEM OF THE A's AND § 5. On the Simplest Eesolvents of the Pkoblem OF THE A's. We will now consider the simplest resolvents of the problem of the A's. It is evident, from what we know of the group of the problem, that we shall have to deal here with resolvents of the fifth and sixth degrees. Our problem will be merely to establish the simplest rational and integral functions respec- tively of the A's, which assume, for the substitutions known to us, five and six values respectively. The principle of transfer- ence developed m § 2 here again serves our purpose : we take the simplest integral functions of \, Xg, which assume for the homo- geneous ikosahedral substitutions Jive or six values, polarise these with respect to \', X^, till a function arises vjhich is symmetrical in the \, \"s, and finally substitute the A's in place of the original X, X"s. As regards the five- valued functions of the Xj^, Xj's, the simplest were : <„(Xj, Xg) = fS-XjO + 2c2-Xj6a2 - 5e'Xi*/.j2 (22) - 5e*''Xj2X2'* - 2e^''\y.rr> + e^''\% W,{\, \) = - £*-XjS + cS-XjfXj - 7£2-Xi6y - TfXjSx^a + Tc^'Xi^X/ - 7e3'Xj2X/ - €2-Xj>./ - fXjS ; to them are added, further, t,^ and t,Wy. Now, polarising tv thrice, W, four times, and introducing the A's, we obtain accordingly as the simplest five-valued function of the A's: (23) b. = 6' (4 Ao^A^ - AjA/) + €2- ( - 2 AoA/ + Aj^} + e^" (2AoAi2 - A/) +(*'{- 4Ao2Ai + A^^Aj), ' = £-(- 4Ao3A2 + SAoAjAj^ - Aj*) + €2" ( - GAo^A/ + AoAi3 + AjA/) + e3- ( - GAo^Ai^ + AoA/ + AjSAj) + €^-(-4Ao3Ai + 3AoAi2A2-A2-); if we want more five-valued functions, we shall take in addition h^ and 8„8 ', corresponding to t,? and t, Wy. The resolvent of the 8„'3 we shall presently discuss more in detail. Of the resolvents of the sixth degree of the ikosahedral equa- tion, we have previously (I. 5, § 15) only considered the one whose roots ^ are given by the formulae : THE JACOBIAN EQUATION. 247 (24) |<^. = 5VV. We obtain from this by our principle of transference the follow- ing roots of a resolvent of the sixth degree of the A's : (25) /^. = 5Ao^ Here, however, wc have exactly the equations of definition of the Jacobian equations of the sixth degree given in II. 1, § 3 ; at most we should have to mark the distinction that here e^ stands where e*' stood before and conversely. But this is merely a difference in the denomination of the roots Zy. If we recur to the formulae which we also communicated, lac. cit., § 5, in describing the Jacobian equations, we learn first that our present magnitudes A, B, C, are exactly identical with those similarly denoted there. We can therefore, without more ado, carry over the form of the Jacobian equation previously communicated : (26) (2 - Af -iA{z-Af+ 105(2 -Af-C(z-A) + {SB" -AC) = 0; the only question is on what basis we shall place it from our present standpoint. The further question arises how far we can replace the problem of the A's by the equation (26), and, in particular, what significance is then to be attributed to our form D. § 6. The General Jacobian Equation of the Sixth Degree. We have already in formula (11) come across the linear functions of the A's whose squares represent the roots z (25) of the Jacobian equation of the sixth degree ; we there saw that these, when equated to zero, represent the polars of the six fundamental points with respect to the conic A = 0, and there- fore certain straight lines which do not themselves perhaps pass through the fundamental points. We can, however, introduce in their place curves which do so ; namely, we recognise at once that the conies : z,-A=0{> = 0, 1, 2, 3,4) all pass through Aj = 0, A2 = 0, and that therefore of the conies: 248 THE PROBLEM OF THE As AND each contains those Jive fundamental ■points whose indices are different from its own. We will now consider tte {z — ^)'s as the actual unknowns. Then the theorem just given permits us to write down at once (paying regard to the definition of B, C, contained in § 3) the coefficients of the corresponding equation, so far as their form is concerned. Let us consider, for example, the sum : ^{z,-A)(z,-A){z,-A) (where the summation extends over all the values of i, k, I, which differ from one another), which will give the third co- efficient of that equation : it must be equal to an invariant form of the sixth degree which vanishes twice for all the fundamental points, and can there only differ from -B by a numerical factor. In this way we obtain directly : (2 - Af + kA{z -Ai^ + lB{z - Af + mC{z - A) + {nW- +j>AC) = 0, where Ic, I, m, n, p, are numerical coefficients which are as yet unknown, but which we determine readily hereafter by returning to the explicit values of the expressions in the A's which pre- sent themselves. The coincidence with formula (26) is obvious. Let us further remark that (26) is in fact transformed into the resolvent of the sixth degree (which we previously established) of the ikosahedral equation, if we put in agreement with (24) and (15) : A = 0, B=-f, C=-H, z = ^. As regards the group of the equation (26) [in the Galois sense], this is determined already by our earlier elucidation of the case ^ = 0, to which we here refer (I. 4, § 15). It is a group of 60 permutations which is holohedrically isomorphous with the group of substitutions of the A's. It must therefore be possible to express the A's rationally by means of our s's. We effect this most simply if we first compute from the equations (25) the squares of the A's and the products in twos, and hence derive the quotients A^ : A^ : A^, and then proceed exactly as in § 4. Here we must manifestly make use of the D in addition to ^, B, C, which alone appear in the coefficients of (26). We can therefore say : THE JACOBIAN EQUATION. 249 The Jacdbian equation (26) is an equivalent of the prollcvi of the A's, if we suppose that, besides its coefficients, D is also given, i.e., (by (16)) : the square-root of a definite integral fmction of A, B, a We now ask how D'^ may be expressed as a rational function of the roots s. To this end we form from (25) the difference of any two ^'s as a function of the A's, and find that this, being a difference of two squares, can, after separation of a constant factor, be also split up into linear factors, such as, by formula (13), also appear in D. We get, for example : 2. - Zj, = {f - f*-) (e'Ai - e^-Aj) ((1 ± n/5) \ + e-'A^ + e^-A,) for v= 1, 2, 3, 4, where + «/5^is to be taken for i^ = 2, 3 ; — Jii for i'=l, 4. If we now multiply all these differences together (each taken once), we obtain on the left-hand side the square root of the discriminant of (26), which we have alread^j^ (II. 1, § 5) denoted by IT. On the right hand, however, the constant factors give i \/5*, the rest just D^, so that therefore : (27) i5==ys -^=71 Here D appears, as we see, in the form of an accessory in-a- tionality, i.e., as an irrational function of the ^'s. This will not be the case if, with Herr Kronecker, we regard not the s's, but the vs's as the unknowns of (26) ; for we can immediately express Ap, Aj, A^, linearly in terms of the vs's. But even then the statement of the problem is not fixed by (26) alone, but the value of D must be given expressly besides. I believe, therefore, that it is not to the purpose to make the Jacobian equations of the sixth degree the keystone of the theory, but that it is better to begin, as we have done, with the problem of the A's as such. § 7. Brioschi's Resolvent. We follow yet further the connection of our considerations with the developments of Brioschi and Kronecker by now studying, first of all, that simplest resolvent of the fifth degree whose roots are the expressions 8^ (23). This must give 250 THE PROBLEM OF THE A's AND exactly Brioschi'a resolvent, of which we gave an account in II. 1, § 5. For the BJs are completely identical, as an actual comparison teaches ns, with what was then denoted [formula (22] by x^. In order to compute our equation of the fifth degree, we first inquire again about the geometrical significance of the BJb. We remark at the outset that all the SJa vanish for Aj = 0, A2=0. They therefore represent, when equated to zero, curves of the third order which pass through all the fundamental points. But more than this : the product of the BJs must be, as an invariant form of the fifteenth degree in the A's, identical with D save as to a factor, while D = represents, as we know, the fifteen connecting lines of the six fundamental points. Hence each of the 8„'s, when equated to zero, represents three straight lines, which taken together contain the whole set of fun- damental points. We verify accordingly the following decom- position : (28) 6, = (.4^A, - .^A^) . ((1 + J 5) \ + .*"Ai + ."A^) .((1- V5)A„ + ^''A, + e-A,). We conclude therefrom that the product h^h^h^^h^ is actually identical with D (not merely to a factor pres). As regards the other symmetric functions of the 8's, we have in any case : for there are no invariant forms of the third or ninth degree. We further conclude from the relation of S to the fundamental points that : 232 = 7,-5, 2&*^l& + mAC, understanding by k, I, on, appropriate numerical factors. On determining the latter we have finally : (29) d^ + lOB.d3 + 5{9B^-AC)d-D^O, agreeing with Brioschi,* and agreeing further with the special formula which we derived in I. 4, § 11, on the supposition that A = 0. The discriminant of (29) is of course a complete square. * In Briosohi's memoir somewhat different numerical coefficients were origi- nally given, but these were afterwards rectified by Herr Joubert : "Sur I'^qua- tion du sixifeme degr^," Comptes Rendus, t. 64 (1867, 1); see in particular pp. 1-237-1240. THE yACOBIAN EQUATION. 251 There is no difficulty in computing the product J[ | (S^ — S^) as an integral function of A, B, C. For ^ = it will become -25^/5.C^byI. 4, §14. The equation (29) is necessarily the more interesting because it represents, in the sense of our previous terminology, the general diagonal equation of the fifth degree. To express it geometri- cally, we can say that the formulae (23) for Z,, inasmuch as they satisfy identically the relation 5'S = 0, 58^ = 0, give a single- valued representation of the diagonal surface on the plane A. This representation is a special case of that well-known one which was given * by Clebsch and Cremona for general surfaces of the third order, and which Clebsch has studied for the diagonal surface just in the form here in question.-f For to the plane sections of the diagonal surface correspond in general, in virtue of (23), such curves of the third order as intersect one another in the six fundamental points of the plane which now become the fundamental points of the representation. Here the inter- section of the diagonal surface with the principal surface is represented by ^=0 (as follows from (29)), while the curves ^ = 0, C=0, taken together represent those two twisted curves of the sixth order on the diagonal surface which are the geometrical locus of points with the pentahedral co-ordinates <„ (II. 3, § 4). This is in accordance with the fact that we have, in § 3 of the present chapter, found the deficiency p of the curves ^=0, ^ = 0, C=0, equal to 4, 0, 0. § 8. Preliminary Remarks on the National Transformation OF OUR Problem. Of the researches mentioned above relating to Jacobian equa- tions of the sixth degree, those still remain which relate to the problem : from a first Jacobian equation of the sixth degree to establish a second by a transformation rational in the Vz's and as general as possible. I will expound these researches from * Cf. Salmon-Fiedler, "Analytische Geometrie des Raumes,'' 3d edition, 1879-80. t Viz., in the memoir just cited : " Ueber die Anwendung der quadratischen Substitution auf die Gleicbungen 5. Grades," Math. Annalen, Bd. iv. (1871). 252 THE PROBLEM OF THE A's AND our own standpoint, without entering further into the historical relations thereof. Our object is to determine three magnitudes, Bq, Bj, Bj, in as general a manner as possible, as rational homo- geneous functions of the \, \, A^, in such wise that they them- selves undergo the linear suistitutions 0/ § 2 when we subject Ag, Aj, Ag, to the same* Our requirement, be it understood, by no means requires that the individual substitution of the B's should be identical with that of the A's ; it is only necessary that the totality of the sub- stitution should be mutually coincident. We know, so far, two possibilities of attaining such coincidence : first, by making the substitutions of the B's actually identical with those of the A's; secondly, by allowing them to proceed from the substitutions of the A's on writing e^ in place of e throughout ; in the first case we speak of cogredient, in the second case of contragredient variables. In the next paragraph but one I shall show how we can thus arrive at a separation of the two cases a priori, and that, besides them, no others possessing individual importance can exist. Meanwhile let us take our cases as given empiri- cally, and ask how they are to be substantiated by definite formulfE. It will be to the purpose to first deal with the corresponding statement of the problem in the domain of binary variables, where we came repeatedly into contact with them in our earlier chapters. Let k^, k^, be homogeneous rational, not necessarily integral, functions of Xj, X2 : (30) K-j = (^1 (Xj, ^2), K, = <^2 (Xj, x^), we require so to determine ^j, ^2> '^i'^'t '^n '^2' ^'^® either cogredi- ently or contragrediently transformed when \j, Xgi ^^^ subjected to the homogeneous ikosahedral substitutions. To this end we construct the form, binary in two sets of variables : (31) F(^^i, \ ; f 1, f'2) = H- 't'2 {\' ^2) - H- '^\ i\i \)- This evidently remains invariant if we subject X^, \, to the ori- ginal ikosahedral substitutions, /Aj, fi^, to the , co-ordinated ones * Our demand for homogeneous functions straight ofi is, one may say, an un- nt'cessary restriction, and one which we can afterwards rumove, but which we w ill retain in our exposition iu order to be able to employ our geometrical phrase- olu-y. THE JACOBIAN EQUATION. 253 (cogredient or contragredient) ; for it is equal to jm-^k^ — fi^K-^ ; and /4j, fi^, and /Cj, k^, undergo in each case identical substitu- tions of determinant unity. Conversely, if we have a form in the \, /x's invariant in this sense, and which is linear in the fj,'s and homogeneous in \, \. then : (32) ^,= -f, ., = ^ will be a solution of the problem proposed. It simply comes to this, therefore : to establish all invariant forms F. Now let us observe the following facts. If we have found two systems of solution of (30) «j, k^ ; k^, k^, the determinant K^K^ — K^^ remains invariant for all the ikosahedral substitu- tions. But this is equal to the functional determinant of the corresponding forms F, F' : bF bF Vi ^Fa F' bF' 6/ii ^f2 and this, therefore, as a rational function of X^, \i must be a rational function of the ikosahedral forms f(\, Xj)) ^(\^ ^2)) T{k-^, Xj). I will now assume that we know some two of the required forms F.^, i^g, with a non-evanescent functional deter- minant. Then, if we apply the identity : F bF, 0/^1 =0, bF^ */*2 it follows, from the theorem just established, that each of the forms we seek is compounded of i^j, F2, in the following form : F ^1 bF bF bF, bfi, bj\ bn2 *M2 (33) F=R,-F, + Ii,.F„ where E-^, B^, are rational functions of/(Xj, \^, S^(\> \), T (Xj, Xj). But, conversely, if we assume B.^, B^, to be rational functions of this kind, and then only take account of the rule that F is to be homogeneous in X^, Xg, F will be a form of the kind required. Hence (33) contains in general the solution of 2 54 THE PROBLEM OF THE A's AND our problem, provided only we regard two of our forms F.^, F^, as known. But this supposition is, in fact, admissible both in the contragredient and the cogredient case. Indeed, we know in both cases the lowest forms F^, F^, i.e., those whose degree in Xj, X,, is as low as possible. In the contragredient case these are the two forms iVj, ilfj, which we always employed in the preceding chapter : (34) \f^ = M, = ^, (\i3 _ 39x^8x^5 _ 26 V V°) ( +^2(26VV-39VV-V'). while in the cogredient case we have the two following : ^' = 6^: >'^^ b\,- >"'■ Thus the question we raised is completely solved, so far as the domain of binary forms is concerned.* § 9. Accomplishment of the Rational Tkansformation. Returning now to the A's, we can begin in their case with a step which is analogous to the transition from (30) to (31) ; in other words, instead of seeking elements B^, Bj^, B„ which are covariant to A^, A^^, A^, in the one sense or the other, we seek an invariant which contains simultaneously both sets of vari- ables. The feasibility of this is, geometrically speaking, founded on the fact that an invariable conic : Bo''+BiB2 = lies in the plane B, and that, in respect to this conic, to every point Bg, Bj, Bg, there is co-ordinated as a covariant a certain straight line, to wit, the corresponding polar : 2Bo.Ao'+B,.A/+Bi.A2' = 0.t If, therefore, the following formulm : (36) Bo = <^o (Ai, A2, A3), Bi = <^i (Ao, A^, A^), B^ = 4>2 (K K ^2) * As regards the contragredient case, we have already become acquainted, in formula (25) of II. 3, § 4, with a particular case which is included in this solution. + Here Ao', Aj', A2', denote the current co-ordinates. THE yACOBIAN EQUATION. 255 co-ordinate the B's to the A's either as cogredients or as conira- gredicnts, the form derived from tliem : (37) F (Ao, A„ A2 ; Ao', A/, A^') = 2<^o . Ao' + ., . \' + , . K, will he invariant, provided we effect the same mhstitutions on the A"s as on the B's. Conversely, provided F is an invariant in the sense explained : are formulae of the nature which we are seeking. We now remark that every F admits of being composed with three such ^'s, which are linearly independent, in the form : (39) F=B,F^ + Il,F^ + R^F„ where H^, R^, R^, are rational functions of the invariant forms, which depend only on A^, Aj, A^, i.e., rational functions of ^, B, C, D. Conversely, if we take R.^, R^, R^, as rational functions of this kind, we shall always obtain from (39) a form F of the nature we desire, where we have it in our power, if we attach importance thereto, to make F a homogeneous function of A^, A^, Aj, Everything is therefore reduced to finding, in two cases, three forms F-^, F^, F^, of as low degree in the A's as possible. In the case of cogredients we solve this problem directly by the construction of polars, a process to which we subject the lowest invariant forms, which only contain A's, i.e., A, B, G. We shall put, namely : F =2A A' + A A' + A A' 6B_ bB 6B (40) r^~6A,-'^<'^hA^-'^^^bA,-'^'' \F -^^ A'+-- A'+-- A In the case of contragredients, on the other hand, we again recur to the principle of transference of § 2. We shall first obtain three forms : invariant for contragredient ikosahedral substitutions, which are of even degree 2n, of the second degree in the /t's, and of the 256 THE PROBLEM OF THE A's AND lowest possible degree in the X's. Then we shall polarise these Si's 7i-times with respect to X, introducing X' by the operation, and once with respect to fi, introducing /x', and shall finally replace the symmetric functions of the X, X"s by the A's, those of the fi, fx,"a by the A"s, writing therefore : (41) Ao = - S (^'-2' + '^2'^l')> \ = ^2^'. A2 = XjXj' ; Aq' = - 9 (f^if^-z + .«2/*^'). A/ = ^2/^2'. A/ = - fj-if^i'." The forms fi which are here most suited to our purpose we can borrow from the data of Herr Gordan previously cited. As the /2j, we choose the form t, which we have communicated in § 11 of the preceding chapter (formula (60)) : (42) nj = ;.i2 ( - x/ - 3x,x/) + 10^1^, . Xj3x,3 + ^^2 i^s\% - x/). In order, then, to obtain 12^, we construct the functional deter- minant of the ground-form a noted in that chapter, and the iVj similarly employed just now : ba *M2 V2 We thus obtain : (43) "2 = /ij' ( - 10\%- + 20Xj3x,7) - 2/XiMo ( - \^° + U\^\^ + X,") + ^i./(-20XiV-10XiV)- Finally, we bring forward as the 12^ the square of iVj : (44) n^ = [mi ( - 7^%' - V) + ft (\' - 7>.f x/)]l Now, applying our process of transformation first to /2j, there arises — disregarding a numerical factor — the following as the simplest form i^j, of the third degree in A^, Aj, A^ : (45) F, = 2Ao'(2Ao3 - SApAjA,) - Ai'(3AoA,= + A^^) _ A^XSAoA^^ + A,^). We now treat iJg (43) in a similar manner, but subtract from * Of course we could also proceed in just the same way in the case of co- gredients ; we should not, however, obtain any results different from those now CDmniunioated, and should only have to repeat once more the process of polarisa- tion which led us above to A, B, C, D. THE yACOBIAN EQUATION. 257 the result, for the sake of simplification, an appropriate multiple of A ■ F^. Thus we have : (46) F, = 2Ao' ( - SAo^AjA^ + eAoAj^A," - A,^ - A^^) + Ai' (leAo^A,^ - SAo^Ai' - 4AoAiA/ + 2Ai*A2) + a; (leAo'Aj^ - SAo^A/ - iAoAi^Aj + 2A1A2*). We finally deal with 12^ (44), and obtain, after subtracting proper multiples of A^ ■ F^ and A- F^: (47) F^ = 2Ao' (32Ao3Ai2A22 _ ^A,^ (A/ + A/) - IGAoAi^A./ + 3AiA2(Ai^ + A/)) + Ai' ( - 32Ao5A22 + 48Ao*Ai3 - 32Ao3Ai A/ - i\^A*K, + 14AoAi2A2*-3Aj5A/-A/) + A/ ( - 32Ao5Ai2 + 48Ao*A/ - 2,2\^A^^A, - 4Ao=AiA/ + 14AoAi^A/ - SAjSAjS _ A/). On introducing the F^, F^, F^, thus obtained in (39), and through this in (38), our task is completely accomplished in the contra- gredient case also. § 10. Group-Theory Significance of Cogredience and contragkedience. We now return to the group-theory question, to which we were led at the beginning of § 8. The linear substitutions of the B's are, at all events, holohedrically isomorphous with those of the A's ; we have finally to deal with the problem of investi- gating in how many different ways the group of 60 ikosahedral substitutions : (48) Fo, V, V,, can be co-ordinated to itself in holohedric isomorphism. Two sorts of this co-ordination are given by cogredience and contra- gredience ; we will show that all others are essentially reduced to these. I must state at the outset what rearrangements of (48) will be regarded as non-essential. They are those rearrangements which arise from transformation in the sense previously explained (1. 1, § 2), which, therefore, replace any V^ by {V')'^V^V\ where by V is to be understood any operation of (48). In the applica- tions, namely, which we have to make, we can always regard such a rearrangement as a mere change of the system of 258 THE PROBLEM OF THE A's AND co-ordinates. If we replace the variable z which is subjected to the ikosahedral substitutions (48) by s'= V'(z), (Vy^V^V will appear throughout in place of F^; and similarly if we regard the V^'s as the ternary substitutions of A^, Aj, A^. With the intention of again applying the " principle of trans- formation" just formulated, we now recur to the generation of the ikosahedral group from two operations S and T, of which the first has the period 5, the second the period 2 (I. 1, § 12). We shall have determined the co-ordination which we seek as soon as we declare what operations *S", T', are to correspond to S, T. Here S' will in any case have to possess the period 5. But, by I. 1, § 8, there are in the ikosahedral group 24 opera- tions altogether of period 5, of which 12 are associated with S, the other 12 with ^S*^. If, therefore, in the co-ordination which we are seeking we call to our aid a modification of these by an appropriate transformation of the group, we can in every case put S' equal either to S or S^. If this is done, S' remains unaltered when we replace V^ in general by S'^V^S' (v=0, 1, 2, 3, 4). Consider now the fifteen operations of period 2 which are contained in (48). If we choose v properly in the trans- formation just mentioned, we can always reduce an individual operation of period 2 to one of the three following : T, TU, U, where U is defined as in 1. 1, § 8 (compare I. 2, § 6). If, there- fore, we have disposed S' in the manner just mentioned, it is suffi- cient to make T' equal to one of the three operations T, TU, U. Compare now the rules of periodicity in I. 2, § 6. In accord- ance with them, ST has the period 3, therefore S'T' must also have the period 3. But now we find in the same place for ST, STU, SU, S^'T, S^TU, S^U respectively, the periods 3, 5, 2, 5, 3, 2 proclaimed. Hence S'T can only be either ST or S^TU. There remain, therefore, but two possibilities : in the one case we put S' = S, T'=T, in the other S' = S'', T'=TU. If we write down the corresponding ikosahedral substitutions, we recognise that S^ and TU emanate from S and T when we change e into 6^. Thus we are, in fact, brought back to just the tioo cases, cogredience and contragredience, as was to be proved. We can evidently repeat for every group the question which THE y A GOBI AN EQUATION. 259 is thus answered for the case of the ikosahedron. If, then, a form-problem is proposed which belongs to a group already investigated, we can demand algebraical developments corre- sponding to those given in §§ 8, 9. I will not enter here on a general exposition of this, which would lead us beyond our sub- ject (see, however, I. 5, § 5). I will only remark that the case of cogredience (which, of course, always exists) can always be solved by the construction of polars, when among the invariants of the form-problem there is one of the second degree. This occurs especially in those form-problems of which the variables Xq, Xj^ . . . x„_i are simply permuted, and which are therefore represented by equations of the n"" degree with unconditioned coefiBcients. If for these we employ the invariant Sx^ in just the same way as we applied the conic B^^-l-BjBg just now (§ 9), we are in a position to make the diflferential coeflScients b 6 b W *^l' ' ' ' *^n-i covariant to cCq, x^. . . a;„_i, where by <^ is to be understood any form which is invariant with respect to the per- mutations of the group. We are evidently led back, as a consequence of this method, to exactly the transformation of Tschirnhaus when we take into consideration, in particular, as the functions ^ the sums of powers of the a;'s. The old process of Tschirnhaus is therefore, together with formulae (38), embraced by a general method relating to form-problems of a certain class. Compare with this what was said in II. 2, § 7, on the co-ordination of points and planes.* § 11. Introductory to the Solution of our Problem. We will maintain for a moment the analogy with the Tschimhansian transformation, and accordingly consider the coefficients i?^, B^, -B3, in (39) as undetermined magnitudes. If we then compute for the corresponding B,,, B^, Bj, the expres- sion Bq^-I- BjBg, we obtain a quadratic form of these magnitudes * We can generalise somewhat the remark in the text. In order that the con- struction of polars may aid in the attainment of our object, it is not necessary that an invariant form quadratic in the re's should exist ; the presence of an invariant form bilinear in the x, x"s is sufficient. In this sense the formulae (35) come under this head, for in their case such a bilinear invariant is forthcoming in the determinant (Xo/ii - XjMs)- 26o THE PROBLEM OF THE A's AND which we can reduce to zero by many different assumptions of Mg, iJj, i?2. We can then, however, as we know, determine Bq, Bj, Bj, directly by means of an ikosahedral equation. This being done, we again apply the formulae (39) or (38), except that we interchange the letters A and 8, and therefore express Ag, Aj, Aj, in terms of B,,, Bj, Bj. The coefficients i?j, B^, R.^, are then necessarily rational functions of the original A, B, C, D, and those irrationalities which we may have introduced in making 8^^+ 6162=0 ; the original problem of the A's is there- fore solved through the intervention of these irrationalities and the ikosahedral equation appropriate to the B's.* I have only explained the general process in order to allow the applicability of formula (39) to come to light. The course which we will now pursue in order to solve the problem of the A's, i.e., to reduce it to an ikosahedral equation, is a much simpler one. We bad : (49) 2 A„ = - (XjX^' + XjX/), Ai = \\;, A2 = - \\; ; v:e will now attempt the solution by supposing the ikosahedral equation constructed on which depends the -^ or -/ respectively, ^2 ^2 which here occurs. Geometrically speaking, this means that we seek to determine the point A by means of one of the two points on A = in which a tangent from A to the conic A meets it, while the general method just sketched — though here we suppose the functions in question as homogeneous functions of Ag, Aj, A^ — co-ordinates to the point A any one covariant point lying on A = Q, and then considers its co-ordinates B^, Bj, Bg, determined not merely relatively, but absolutely. The analogy of our statement of the question with that which we have dealt with in the preceding chapter, according to Herr Gordan's plan, is obvious. In both cases we are concerned, as we know, with a form-problem of which the variables are bilinear forms of two series of binary variables which are simul- taneously subjected to the ikosahedral substitutions; in both we seek the solution by returning to the ikosahedral equation on which the variables of the one series (in so far as their ratios • We have already mooted the same point (when speaking of the Jacobian equations of the sixth degree) in II. 1, § 6. THE yACODIAN EQUATIO.X. 261 are concerned) depend. We shall accordingly be able to follow precisely tbe course of ideas which was developed in §§ 6-11 of the preceding chapter ; the individual steps are so simple that it appears superfluous to build up in detail the several results. We began with enumerating these homogeneous integral functions of \, \, and \', Xg'i which remain unaltered for the simultaneous (here cogredient) ikosahedral substitutions of these magnitudes (invariant forms). We have placed side by side in formula (35) the two simplest forms linear in the \"s ; they were the following two : ( W - K\' = J A, (where the computed value of the first form is declared, and the letter P is henceforth introduced for the sake of brevity). To these belong further, as we remarked in § 2, all other forms which arise from f{\, \^, If(\, \), T{\, \) by polarisation with respect to X,', \'.* Our A, £, C, D, the " known " mag- nitudes of the form-problems, are those combinations of the forms here mentioned which are symmetrical in the X, X"s. We consider now generally the interchange of the X, X"s, i.e., the replacement of Xj, \^ by Xj', Xg', and conversely. If an invariant form remains unaltered for the interchange of X, X', it is an integral function of ^, B, C, D ; if, on the other hand, it changes its sign on permutation, it is the product of J A (50) and such an integral function. If an invariant is only of the same degree in the X, X"s, it can always be put into the follow- ing form : (51) F{\, \ ; Xj', A,') = G {A, B, C, D) + JJ . H(A, B, C, D), where the integral functions (?, //, are defined by means of the following equations : ,5^. r _ 2G = F{X„ X,; Xj', \;) + F(X^', -A.: ; \, \}, ^ ''^ X^Ja. h^F{\, X, ■ \', \') - F iW x; ; \, X,).- The general course of our method of solution will now be as follows. We have first to construct the ikosahedral equation : * I do not further press the point that with the forms thus enumerated the entire system of the iuvariauts here coming binder cousideratiuu is exhuuated. 262 THE PROBLEM OF THE A's AND on whicli \ : \ depends, and then to express the invariant P (50) in terms of \, \i "^^ and t^e known magnitudes. Both steps are accomplished by appropriate applications of formulae (51), (52). We then consider the formulas (50) as linear equations for the determination of X^, \' : the final formulae for A^, Aj^, Aj, which we sought are given on intro- ducing in (49) the values which are found. Here A^, A^, A^, necessarily appear as particular linear combinations of the linear invariants ^A and P. § 12. Corresponding Formula. The formulae which are required in virtue of the general method just given are now to be developed so far as appears desirable for giving preciseness to the course of our ideas. I will here again (as in the preceding chapter) denote the forms originally given us by the index 1, the others which arise from them by interchange of the variables X, X', by the index 2. Higher indices may proclaim the degree of integral functions of the arguments adopted in each case, on the understanding that these arguments are considered as functions of Ag, Aj, A,. We begin with the computation of Z (53), or, as we now say, of Zy We have evidently, in order : H,^___Hlf, ^ Hlfl + Hll,^) + (H,^fl-nif,^ _ r;,(, {A, B,C)+ JA.D.G^ , (A, B, C) 3456 [^12 {A, B, C)f Besides (51) and (52), I have here made use of the fact that, among the given magnitudes A, B, C, B, only D is of uneven degree in the A's, and also that Ifl is an integral function of A, B, C. The integral functions G^^, G^, G^ of A, B, C, remain to be estimated by recurring to the explicit values of the magni- tudes in Ap, Aj, Aj, which come under consideration. The com- THE yACOBIAN EQUATION. 263 putation in question is, of course, somewhat formidable ; I omit it, because it furnishes nothing of special interest. We now turn to the computation of P, or rather of Pj. The form Pj is of the first dimension in the \"s, of the eleventh in the X's ; if we wish to employ a process like the one applied to Zj, we shall have to first afiect P^ with such factors dependent only on \j, Xj, that the aggregate which arises is uniformly of the first dimension in the X, Vs. We put accordingly : (55) Pi = ^y^^ -' 1 and then have, in order : (56) '1 T T T -'1 -'l-'2 _ (P ^g^r, + p,g,r i ) + (Piffi?', - p,g,r, •1TJ^_ D . G,, {A, B, C)+ sjA- G,, (A, B, C) 1T^,{A,B,C) where the integral functions G-^^, Ggf^, F^ remain to be evaluated. On substituting, we have ,^ T, D ■ G,, (A, B, C)+ JA. G,, (A, B, C) ^'^ '~K 2r,,{A,B,C) We now seek, as we suggested, to obtain the \', Xg' from J A and Pj. The formulae which arise run simply : '■1 = - V4 • 1-07 + J- J fof ' ( V= + ^/I■]§^+A■ 12/1 Comparing it with (49) we have finally : '2^=-^/I-2P,.^J, x'f^ (59) 12/1' I A.= -VZ.-,-^-P, T2A w here we suppose for Pj the value (57) introduced. 264 THE PROBLEM OF THE A's, ETC. We can in many respects modify the method of solution thus given if we like to take up once more the course of development adopted in the preceding chapter. Substitute, for example, in (59), instead of P^ the magnitude /Dj (55), so that the A's depend only on J A, p^, and X^ : Xg, then compute the corresponding problem of the A's regarding these three magnitudes as arbi- trarily given, and compare it with the proposed problem. We thus obtain for p^ and ^^ (53) determining equations which can be applied to the actual computation of them. We can also, as we did in the preceding chapter, interpret geometrically each step of the method of solution. Leaving all these things to the reader, I emphasise, in conclusion, the appearance of J A. In the sense of our previous mode of expression, this is an accessory irrationality, i.e., one which is not rational in the magnitudes Ap, Aj, Ag which are to be computed.* We shall soon see that an irrationality of this kind is, in fact, indispensable if we want to reduce the problem of the A's to an ikosahedral equation. * In an analogous sense, the notion of the accessory irrationality is transferred generally to form problems. ( 26s ) CHAPTER Y. THE GENERAL EQUATION OF THE FIFTH DEGREE. § 1. Formulation of Two Methods of Solution. TuRNLNG now to the general equation of the fifth degree, let us attack forthwith the actual problem of solution.* We are in principle concerned with the construction, from five magni- tudes, Xf^, aij, . . . x^, which are subject to the single condition 2'ic = 0, of a function {X(„x-^, . . . x^) = X, which undergoes ikosa- hedral substitutions for the even permutations of the a;'s. How we shall afterwards represent the individual a;'s in terms of X is a question in itself which at first we regard as a secondary one. Limiting ourselves first to the main question, let us take a geo- metrical interpretation as our basis ; we regard ajg : iCj : . . . x^, as we did above, as the co-ordinates of a point in space, \ as the parameter of a generator of the first kind on the principal sur- face of the second degree 5*^:^ = 0. Our problem then becomes: to any point x in space to co-ordinate by appropriate construc- tion a generator \ in a covariant manner, i.e., to co-ordinate in such wise that the relation between the point and the generator remains unaltered when both are simultaneously subjected to the even collineations. A first solution of this problem arises of its own accord, so to say, on the ground of the developments already given. Namely, vje shall at the outset exhibit in covariant relation to the point x a point y of the principal surface, and then take as generator \ the generator passing through y. Therefore, to characterise at once the algebraical treatment of the general equation of the fifth degree which arises from this, we shall transform the general * The developments given in the following pages are contained in their general features in my oft-cited works in Bd. 12, 14, 15, of the Mathematische Annalen, but they are here for the first time e.xpounded in a connected form. 266 THE GENERAL EQUATION OF equation of the fifth degree by an appropriate Tschirnhausian transformation into a principal equation, and then solve this in accordance with tJie method expounded in the third chapter of the present part. The Tschirnhausian transformation which is required in the process now described has been mentioned in greater detail in II. 2, § 6, and there formulated in the following manner : we first co-ordinated to the point x a straight line in space which joins two rational points covariant to x, and then chose as our point y one of the points of intersection which this straight line has in common with the principal surface. Here, generally speak- ing, an accessory square root would be necessary for separating the two points of intersection. If we wish to express ourselves briefly, we can even put aside the point y in our description of this construction. Our object is then simply to employ one of the two generators of the first kind on the principal surface, which meet a straight ILue which is covariant to x. The accessory square root depends on the fact that alongside of a first gene- rator of this kind which we call \, there is always a second associated with \, which we will denote by \' for a moment. Expressing ourselves thus, we recognise the possibility of still further postponing the use of the accessory irrationality. In- stead of seeking at once the ikosahedral equation on which \ depends, we shall first establish the equation system by which the symmetric functions of X, V, are determined, and not till later deduce from this equation system the aforesaid ikosahedral equa- tion. But this is manifestly the same as saying that we return to the developments of the fourth chapter which we have just concluded. In fact, our \, X,', are cogredient variables ; the equation system of which we speak is therefore an equation system of the A's, in the treatment of which we shall, moreover, be led at once, as we shall see, to the homogeneous arrangement, i.e., to ihe form-problem of the A's. At the same time, the ikosa- hedral equation on which \ depends is the same as we should anyhow use in the solution of the problem of the A's. We therefore find a second method of solution of the general equation of the fifth degree, in which we turn to accoimt the developments of II. 4, exactly in the same way as we do those of II. Z for the first method of solution. THE FIFTH DEGREE. 267 For tlie rest, the formulation which, we have just established for the second method of solution is unnecessarily precise. Recalling the considerations which we have given in II. 2, § 9, we recognise that we can co-ordinate to the point x, instead of a straight line, a general linear complex in order to effect the second method. The generators \, X,', are then those two which helong to this linear complex. The explicit formulae, which we shall establish later with a view to giving exactness to the second method, remain undisturbed by this generalisation ; we shall therefore only quite cursorily return to the special formu- lation which we just now began. We have now a twofold task in the following paragraphs. In the first place, we shall have to establish the more exact formulae which correspond to the two methods of solution, the feasibility of which we ascertained ; and then we will bring the totality of those researches on which we reported in II. 1 into conformity with our own reflexions. In this respect the relationship of our first method of solution with that of Bring and of our second method with that of Kronecker is evident at the outset. By using a theorem which we previously established (I. 2, § 8) concerning the ikosahedral substitutions, we then succeed in proving also that fundamental proposition of the Kronecker theory to which we referred in II. 1,§7. § 2. Accomplishment of our First Method. In order to give exactness to our first method, let (1) vfi + a3? + hx'^ + cx + d=0 be the given equation of the fifth degree (in which we have taken ^(a;) = 0). We then further put, in accordance with II. 2, §5: (2) ijv =P ■ xj" + q ■ x}"^ + r . a;^''' + s . x,}*', where x*^ = xj' — _ 2'x', and compute Sy'^- This is a homogeneous integral function of the second degree of ^, q, r, s : (3) * {P, ?. '•. «). of which the coefficients are symmetrical integral functions of the x's, and therefore integral functions of the coefficients 268 THE GENERAL EQUATION OF a, h, c, d, occurring in (1). We wish to find a solution system, of the eqiuition $=0 which remains unaltered for the even permv/- tations of the x's. Let us first remark that the p, q, r, s, required cannot possibly be equal to rational functions of Xg, x^^, . . . x^. This follows from the proof, to be presently brought forward, according to which the use of an accessory irrationality, at least, therefore, an accessory square root, is indispensable for carrying out our method.* We return the more readily to the geometrical construction with the covariant straight lines which we described just now, and for which we have given the necessary formulae inll. 2, §6. Let: Jr-^, Wj, xij, Oj ', Jr^t '^s' -^2' "^2 be two series of four magnitudes which depend rationally on the x's, and in such wise that they are not altered for the even permutations of the x's^ and which are therefore rational func- tions of the coefficients a, h, c, d, of (1) and of the square root of the corresponding discriminant. We then put in (2) as before : (•i) i' = />i-Pi + P2A. 1 = P\Qi + Po.Q-2, r = p^R^ + p^R.^, s^p^S^ + p.^S^. By this means $ [formula (3)] is transformed into a binary quadratic form of the p^, p.^'a, the coefficients of which are rational functions of the known magnitudes : we put $ = 0, and deter- mine Pj, P2, from the quadratic equation which arises, whereby the proposed accessory square root is introduced. Then let us substitute corresponding values of pj, p^, in (4) and (2) respec- tively, and compute the principal equation which results for the y's, which we will briefly denote as follows : (5) f + 5a,f + (,^y + y = 0. Thus we have made every arrangement necessary for the imme- * Conversely, if we proved the theorem in the text (concerning the irrationality of p, q, r, s) directly, vf& should have a new proof of the necessity of the square root in question. Namely, could an ikosahedral equation be produced from (1) without employing accessory irrationalities, we should be able to construct from this one of the infinitely many corresponding principal resolvents of the fifth degree, and then obtain, by collecting the formulae, a transformation (2) of which the coefficients p, q, r, s, would be rational functions of the x's, unaltered by the even permutations of the x's. THE FIFTH DEGREE. 269 diate application of the developments of II. 3. If we have then computed the roots y^ of (5) by the help of these develop- ments, we find the corresponding x/s by reversing (2). I should like here to make a passing remark with regard to the inversion of the Tschimhausian transformation. It is usually said, and we have also so expressed it farther back (II. 1, § 1), that the x„ required is computed rationally as a common root of the equations (1) and (2) [where now y, is to be regarded as the unknown magnitude]. It is essentially the same, but more in the spirit of the rest of our considerations, if we place opposite formula (2) another explicit one : (6) X, =p' . 2/,"' + q' . 2/;^' + r' . yj^^ + s' . yj\ where yJ''^ = yJ' — -^Sy'' and p, q', r', s', denote rational functions of Pj, P2, a, h, c, d, and of the square root of the discriminant of (1), which are computed by the aid of elementary methods. The determination of the Xy's can then be conceived in this sense, that, geometrically speaking, we derive from the first found point, y = y^^\ three more covariant points, 3/'^', y, y'*\ and then construct the required point x by means of invariant coefficients.* Let us consider that in the method of solution here explained the computation of a;„ from the root \ of the ikosahedral equa- tion finally established is divided into two steps ; we have originally, in II. 3, employed the five-valued functions of X : _ 12/^(X).a>.) _ 12/(X).TF.(X ) """ 3'(X) ' " i/(X) in order to compose the 2/„'s linearly from them, or from v, and UpVy-, we have then represented the point a; as a linear com- bination of y, 2/'^', y, 2/'^'. We can evidently condense these two steps into one : we can compose the point x from four points which are covariants of the generator X. The simplest rational functions of X, which assume on the whole four values for the * The geometrical mode of expression in the text is of course only a counter- part of the algebraical process, when the latter is so specialised that the law of homogeneity is satisfied throughout, i.e., that the ratios of the ys only depend on the ratios of the x's. We ought really to repeat the'same remark in all the follow- ing developments, but for the sake of brevity we shall omit doing so. 270 THE GENERAL EQUATION OF ikosahedral substitutions, are, by what precedes (I. 4), the following : Here Su = Sv = Suv = 0, while on the other hand Sr^O, so that instead of r, we will introduce the combination r„ — zS*?". Then: (7) x^ =y . Uy + q" . Vy + r" . UyVy + s" . yy—K -'' )> where p", q", r", s", are coefficients of the same nature as /. 2', r, s'. I have annexed this new formula of inversion really for the sake of completeness. In fact, it is just this which appears to me to be the peculiar advantage of our first method — that when formulated in the way represented by (6) it is decomposed into two separate parts, of which the first, which is concerned with the connection of the general equation of the fifth degree with the principal equation, has throughout quite an elementary character. We can even consider formula (6) as more simple than formula (7). Namely, if we consider Pj, Q^, . . . E^, S^, in (4) as rationally dependent on a, b, c, d, alone, not on the square root of the corresponding discriminant, then the square root of the discriminant will also be wanting in the coeflScients of (6), while it necessarily appears throughout in the coeflicients of (7), as also in the right-hand side of the ikosahedral equation for \. § 3. Criticism of the Methods of Bring and Hermite. Before going further, we shall compare our first method of solution with the closely related kinds of solution which Bring and Hermite have respectively given. The details which here come under consideration have already been developed in II. 3, §§ 13, 14. iVbw tJiai we come back to these, we wast describe our method as an essential simplification of the Bring method. Bring, too, transforms the given equation of the fifth degree into a principal equation ; he, too, employs the rectilinear gene- rators which lie on the principal surface. But beyond this he comes to an unnecessary complication : in order to obtain a THE FIFTH DEGREE. 271 normal equation with only one parameter, he thinks that a new accessory irrationality has to be introduced by the intervention of an auxiliary equation of the third degree. I therefore main- tain that the original process of Bring may be dispensed with, and must be replaced by our first method, which retains the essen- tial idea of the Bring method. The advance with which we are concerned finds significant expression in the "deficiency" of the figure to be employed for the geometrical interpretation ; the family of rectilinear generators (lying on the principal surface) of the one kind or the other form a manifoldness of deficiency p = ; the deficiency of the Bring curve is equal to 4.* As the crowning point we shall embrace in the critique thus formulated the process of Hermite also : if we wish to apply elliptic functions to the solution of the principal equation of the ffth degree, this is done most simply by using the formula given in I. 5, § 7, for the root of the corresponding ikosahedral equa- tion. Hermite's use of the Bring form can only come under con- sideration thenceforward if, instead of the rational invariant ^, to which the right-hand side of the ikosahedral equation is equal, we employ the corresponding /c^. In fact, we saw in II. 3, § 14, that the cubic auxiliary equation of Bring becomes reducible when we consider k as known. I will also here bring into special prominence the advance which is made by our having deduced directly from the form of the ikosahedral equa- tion the possibility of solving the ikosahedral equation by means of elliptic functions. § 4. Peeparation for our Second Method of Solution. The geometrical opening which we have given for our second method of solution requires us to establish the quadratic equa- tion on which depend the two generators of the first kind on the principal surface which belong to a definite linear complex. We have solved this very problem in II. 2, § 10, for any surface * Starting from the value oi p and the general theory of curves with p=i, we can show (as I cannot here do in detail) that Brirg's cubic auxiliary equation is, in fact, indispensable if we would determine a point of the Bring curve, i.e., employ the trinomial equation f + 5Py + y=0aa normal equation. 272 THE GENERAL EQUATION OF of the second degree, tlwtigh only for the case, we must admit, of a particular system of co-ordinates. We had then taken as the equation of the surface the following : (8) X,X, + X,X3 = 0, and had then defined the parameter \ of the generator of the first kind in the following manner : (9) x= _"?! = '?3 = ^ and, finally, understanding by ^^, y, the co-ordinates of the linear complex, we had obtained the equation : (10) ^42^2 + (^23 - A J W + ^igX/ = 0. I add here at once the corresponding formulae for the generator of the second kind. We found as the defining equation of the parameter fj, : (11) ^=-f = f = lh, -*4 ^1 1^2 and as the corresponding quadratic equation : (12) - A^^^^ + (^23 + ^u) Fi^'a + ^i2M2- = 0. We now recall the method by which we introduced the para- meters \, fi, for the principal equation in particular, in II. 3, §§ 2, 3. This was done in exact agreement with (8), (9), (11), only that instead of Xj, Xg, X„ Jl^, we wrote p^, p^, p^, p^, respectively, where p^ denoted the expression of Lagrange : (13) Pi^ = x^ + €i^. a-j + tS/'. x^ + e^i^ . x^ + e^i^.x^. We can therefore retain equations (10), (12), unaltered, provided we proceed throughout on the basis of Lagrange's system of co- ordinates in dealing with them. In II. 2, § 9, where we co-ordinated to the point x a covariant linear complex in the most general manner, this latter supposi- tion has not been made; the co-ordinates there given for the complex : (14) a,4 = ^c'' "• {xf^r - xf'4"} THE FIFTH DEGREE. 273 [where the c'-" denote any rational functions of the coefficients a, 6, c, d, and the square root of the corresponding discriminant] are related to the fundamental pentahedron in the same way as the point-coordinates x themselves. Our first task is there- fore a transformation of co-ordinates: we must determine the co-ordinates A^y which the complex (14) assumes if we introduce, by means of (13), the expressions p^. To this end I will denote those ^'s which belong to the points a;"*, x""', by p^'\ js""' We then have : (15) p^V"' -p'i¥:'=^ (^-^"^ - ^"^+''^) (4'U"'> - =cN:% i, k where on the right-hand side each combination (i, k) = (k, i) occurs once. We now add the six equations which we obtain in this manner for the difiFerent combinations (Z, 7n) = (m, l), after we have multiplied each by c''™ [formula (41)]. There thus remain on the left side the A^Js required, while on the right side sets of six terms are condensed into the a.^'s (14). Hence the formulas of transformation which we seek run thus: (16) A^y = ^ (£f''+-^- - t-i+f''^^) • «,i. We now introduce the ^^^'s so obtained into (10), (12). I will write the quadratic equations which here arise in the form which we just now established in II. 4 : ^^'> \ A>i2 + 2A>i^2 - A>2^ = 0. We then have : 2 A= +^23-^4= y^('''^'*-^'^''+'"^*-''^")-""' (18) i,k Ai=+^42 = >'(.*"+^-c='+^0-a,,; A, = -^13 =2^(r'+*-e'+=^)-a,,, likewise also : 274 THE GENERAL EQUATION OF (19) i.k I. It i.k § 5. Of the Substitutions of the A, A"s — Definite Formulation. In virtue of the geometrical considerations which we have placed in the foreground, it is manifest that the ratios of the Ag, Aj, Aj, just established undergo exactly the same linear substitutions for the even permutations of x^, Kj, x^, x^, x^, as the ratios of magnitudes established in the preceding chapter, and denoted by the same letters ; it is likewise evident that the ratios of the A"s introduced in (19) behave contragrediently to the ratios of the A's. I say now that this correspondence holds good if we regard, instead of the ratios of the A, A"s, the A, A"s themselves. It would not be hard to prove the accuracy of this assertion on general grounds. We shall presently, § 9, indicate the method of doing so ; meanwhile let us be satisfied with veri- fying its accuracy from the formulae. We have evidently only to take the two operations S, T, into consideration, all the others being composed from these by iteration and combination. First, as regards the even permutations of the x's, we have in II. 3, § 2, introduced for S, T, the following : I J : a^Q = Xqj x-^ = a^2» •'^2 ~ "^i' "^3 ^ "*'4' "^4 ~ ^3* Corresponding to them, we obtain definite permutations of the «jj.'s (14), and, if we take account of these, we have the follow- ing substitutions for the A's defined by (18) : (o ; Aq = Aq, Aj = t*A2, A2 = ^Ag J I'V5-a; = Ao + Ai + A2, T: 'Vo- A/ = 2Ao + (e2 + ca)Ai+(e + £<)A2, l J5 ■ A,' = 2Ao + (€ + c^) Al + (£2 + 63) A2, i.e., exactly the same substitutions as we have given in II. 4, THE FIFTH DEGREE. 275 § 2.* As regards, however, the contragredience of the magni- tudes A' (19) and the A's (18), it is sufficient to remark that the values of the A''s proceed from those of the A's if in the latter we change e into e^ throughout. We now suppose any of the invariant forms of the A, A"s (18), (19) constructed, such as we described in the preceding chapter ; either therefore the expressions A, B, C, D, from the A's alone, or from the A, A"s simultaneously the functions F^, F2, -F3, linear in the A"s, which we have considered in § 9 of the same [see especially formulae (45), (46), (47)]. On introducing for the A, A"s the corresponding values in Xg, Xj, . . . x^, we obtain throughout rational functions of the x's such as do not alter for the even permutations of the x's, and which therefore admit of expression, by the help of elemen- tary methods which we do not carry out, as rational functions of the coefficients a, h, c, d, occurring in (1), and of the square root of the corresponding discriminant. In order to formulate our second method in a definite manner, we at first employ only the problem of the A's, and therefore the values of the magnitudes just mentioned. A, £, C, J). We then follow the developments which we have given in the two concluding paragraphs of the preceding chapter, and construct, after adjoining the accessory irrationality J A, a corresponding ikosa- hedral equation for the determination of \. The only ques- tion which remains is how we will conversely express the roots Xq, x^, . . . x^ by the help of this \. This is to be dealt with in the following paragraph. § 6. The FoRMULiE of Inversion of our Second Method. In order to solve the problem which still remains, no less than three openings present themselves, viz., according as we wish to solve our problem at one stroke or decompose it into two or three steps. In the former case we make immediate use of the formula (7), * The letters Ao', Ax', A2', are employed in (21) in quite a different sense from that of (19) ; as I do not recur hereafter to (21), no misunderstanding will, I hope, arise from this. 276 THE GENERAL EQUATION OF which produce again (laying aside now the accents there em- ployed for (jp, q, r, s) : (22) X,. =p .Uv + q. v^ + r . u-jo^ + s f r^ - -S?-^ j. Here jp, q, r, s, are rational functions of a, h, c, d, of the square root of the corresponding discriminant, and the accessory square root J A. In the second case we first express A^, Aj, A^, as we did in detail in § 12 of the preceding chapter, in terms of the root X. of the ikosahedral equation. We then further bring to our aid the lowest five-valued integral functions of the A's. According to § 5 of the preceding chapter these are : K, V, K\ KK, Here again 5S = 2'S' = 5SS' = 0, while 2'S^ is different from zero, so that for the expression of the x^s we will introduce, instead of the individual h^^'s, the combination (S^°— rS'S^). We have then again formulae of the following kind : (23) a-„ =y .h, + q . ij + r . U,^ - gSS^ + s' . &X, where p', q', r', s', are rational functions of a, b, c, d, and the square root of the discriminant, hut tm longer contain the acces- sory square root J A. Finally, in the third case, we suppose Ag, Aj, Aj, again com- puted first from the root \ of the ikosahedral equation ; but then, instead of seeking the x^'s directly, we first seek the corresponding A^', Aj', A^' (19). We effect this by calculations analogous to those which we previously made on expressing the forms F-^, F^, F^, just mentioned, which depend on the A, A"s, as functions of a, h, c, d, and of the square root of the discriminant, and determining A^', A^', Ag', as unknowns occur- ring linearly. This being done, we seek the simplest possible functions of the A, A"s, which are five- valued and at the same time symmetric in the A, A"s. We find a first function of this kind if we square the y, of II. 3 : y^ = (*- . X,/[xj - €3" . X.JJU.J + £^'' . XjjLt, + «" ■ >^2H-2 THE FIFTH DEGREE. 277 and submit it to the process of transference continually em- ployed in § 2 of the preceding chapter. In this manner there arises a form bilinear in the A, A"s : (24) X' = 2Ao' (f^-Ai + e-A^) + A/ ( - 2€3.'Ao + e^.^j + e^^A^) + Aj' ( - 2£2^Ao - cAi + e^-Aj). As other functions with the same properties we will employ the powers Xv^, ■)(y^, jj;^*, where, however, we must consider that none of the power-sums S^^, 2'x^, J^*' vanish identically. We shall therefore do best to write the formula which corresponds to (22), (23), with an extra term, as follows : (25) x^ = p" . X- + q" ■ -x/ + r" . Xy" + s" ■ Xv^ + ^"• Here p", g", r" , s", t", are again at the outset rational functions of a, b, c, d, and of the square root of the discriminant. More- over, we can bring it to pass that they shall be merely rational functions of a, b, c, d. We have only to then make the c'-^'s in the original opening (14) themselves depend rationally only on a, 6, c, d. I have brought together these data without detailed elabora- tion, because they, so to say, of necessity proceed from the pre- vious developments. The third kind of opening appears to me unquestionably the most effective. Decomposing, as it does, the computation of the xJb into not less than three separate steps, it employs three times over the same elements of the typical exposition with which we have become acquainted under varying forma in the three preceding chapters. § 7. Eelations to Kronecker and Brioschi. Our second method of solution is, as we have often said before, only a "modification and extension of the Kronecker method. In. fact, we have seen in detail in II. 4 that the problem of the A's, in the sense there explained, can be replaced by its simplest resolvent of the sLsth degree, the Jacobian equation. In the details many points of difference certainly present themselves, I will here only call attention to two, of which the second is the more important. We first remark that the way in which Herr Kronecker, 278 THE GENERAL EQUATION OF in his first communication to Hermite,* reduces tlie general Jacobian equation of the sixth degree to the case A = 0, or, as we now say, to an ikosahedral equation, is different from the method applied in the preceding chapter. Herr Kronecker so formulates his method that A^, Aj, A^, contain a parameter v which occurs linearhj, and which is afterwards so determined that Ag2 + AjA2 = ^ becomes zero. We can, of course, combine this idea with our formulae, viz., by providing at the outset the c^'^'s themselves [formula (14)] with a parameter \ occurring linearly. Then, instead of distinguishing by an accessory square root the two generators of the first kind, which the linear com- plex in question for any value of v has in common with the principal surface, we proceed thus : we first make the complex variable in a linear fasciculus, and then fix its position by the condition that it shall contain two coincident generators of the first kind belonging to the principal surface. This condi- tion itself brings with it an accessory square root. I have in what precedes dispensed with the formulation thus pointed out, because it is only applicable if we treat the problem of the A's as a resolvent of the proposed equation of the fifth degree, while I wished to first consider the problem of the A's independently of such connections. We further remark that the general formiilcc which Signor Brioschi has given for the accomplishment of the Kronecker method, formulae of which we gave a detailed account in II. 1, § 6, are throughout different from, our formulae (18). Signor Brioschi employs for the construction of his A^, Aj, A^, six linearly independent magnitudes, m„, m^, . . . u^, while we use twenty magnitudes a^^, between which the relations a^= — ««> Sajit = 2«i;t = subsist. Again we are satisfied with the same magnitudes, ajj., when we wish to take under consideration the A"s alongside of the A's, while Signor Brioschi would have to annex six new magnitudes, ?6„', m^', . . . «/. I will not pursue this comparison, which only concerns the external configuration of the formulae, any further. Let us remark, above all, that our formula: (quite as much as those of Brioschi) are in any case as general as they can le. If, namely, the A, A"s are arbitrarily * See II. ], § 6. THE FIFTH DEGREE. 279 given, we can from them determine conversely the correspond- ing a^s and c'-^'s respectively [formula (14)]. We have only to repeat the transformation of co-ordinates of § 4 in a reverse sense. The calculation in question takes the following form. "We have first, on returning from (18), (19), to the co-ordinates, A^. (16) : (26) j^i3=-A„ ^ A,, = k^, ^ -^14 = ~ Ag - A„', .423 = Ag - Ag'. We then replace the formulae (13) by their reciprocals : (27) 5x, = €-' . pj + e-« . p^ + c-'* ■ iPg + e-« . p^. Hence : 25 {xfx'r^ - 4"^^' = ^(r'''-'* - e-"-*) (^»^i;"> -^<'>p W), where the summation on the right-hand side extends over all combinations {fi, v) = {v, fi), and now, on multiplying the indi- vidual equation by c'-" and adding the several terms for (I, m) = (m, I) : (28) 25aii = ^(£ ■fii-vk __ £-vi-^t\ ^ ^ /Xl') which is the formula we sought. I should like, in conclusion, to formulate concisely once more the geometrical idea which lies at the root of our treatment of the Kronecker method, and which probably possesses far- reaching significance. The first thing is, that we substitute in general for the point x a linear complex, considering, there- fore, instead of the equation of the fifth degree, an equation of the 20th degree whose roots a^. satisfy the oft-mentioned relations a«;= — «fci, San, = Sa.ik = 0. The second thing is, that t k we refer this complex by means of (18), (19), to a new system of co-ordinates. I will not enter* into any details concerning the significance of the A, A"s, but only remark that the first of * Consult my essay in the second volume of the Annalen (1869) : "Die allge- meine lineare Transformation der Liniencoordinaten." Consider, in particular, that the linear complex becomes a special one, i.e., a straight line, when Ao^ + AjAa = Ao'- + Ai' A/. 2 8o THE GENERAL EQUATION OF the two equations (17) vaniahes identically when all the A's are equal to zero, the second when all the A"s are zero. Therefore, for the generators of the first hind, f^^ = ^^ = P\.^ = Q , for the gene- rators of the second kind, A„' = Aj' = A2' = 0. What is the object of this transformation of co-ordinates? By its means we are enabled to replace the equation of the 20th degree for the a^'a by the forin-prohleni of the A's or the A"s. In fact, we have seen that, for the 60 even collineations of space, A^, A^, Ag, and likewise Ap', Aj', Ag', are linearly substituted in their own right, and therefore as ternary forms. Let us now remark that we could have premised, a priori, this property of our geometrical conception. Namely, for the even collineations of space each of the two systems of rectilinear generators becomes transformed, as we know, into itself. Hence of necessity the two threefold families of linear complexes to which these systems of generators respectively belong are also transformed into themselves for these collineations. But from this follows directly the property of the A, A"s described, provided we further postulate that to every collineation corresponds a linear transformation of the line co- ordinates. Tlie possibility of reducing our eqioation of the a^j^s to a ternary form-problem thus appears as an imnudiate outcome of the elementary intuitions of line geometry. This is the particular point of view under which I should like to see the second method considered. § 8. Comparison of our Two Methods. The two methods for the solution of the equation of the fifth degree which we have contrasted with one another are, never- theless, as follows from the considerations of § 1 of the present chapter, very intimately related : we will here show that it is only in non-essential points that they differ, inasmuch as every ikosahedral equation which is co-ordinated to a proposed equa- tion of the fifth degree by virtue of the one method can always be deduced by means of the other method. The passage in this sense, from the first method to the second, is immediately evident. In order to co-ordinate a point y of the principal surface as a covariant to the point x, we have just now (§ 2) constructed first a straight line covariant to x, and have THE FIFTH DEGREE. 281 tli,eii intersected witli this the principal surface. We can now start from this very line as the special linear complex of the second method : we have only to compute the corresponding co-ordinates ttjj. If we, then, construct the corresponding problem of the A's, one of the two ikosahedral equations by which we can solve this problem will immediately become identical with the ikosa- hedral equation to which the determination of the 2/„'s leads. The converse of this argument is not much more complicated. We assume that we have by means of our second method co- ordinated to the equation of the fifth degree an ikosahedral equation, and therefore to the point x a generator X of the principal surface. Then we can always find in a rational inanner (and this in many different ways) a point y which lies on the generator X: we need only, for example, make the yjs proportional to the W^{\)'a or to the other expressions which occur in the principal resolvent of the ikosahedral equation. But this point is co-ordinated to the point x in any case as a covariant ; we have therefore at once a Tschirnhaitsian transfor- mation which co-ordinates to the point x a point y on the principal surface. If we now make this Tschimhausian transformation the basis of our second method, we return of course to the initial ikosahedral equation. In this sense we can say that in reality only one solution of the equation of the fifth degree is found. The difference between the two methods which we proposed lies only in the order of the individual steps. In the first method we give prominence to the accessory square root, in the second we do not introduce it till after separating the two systems of gene- rators. Against this, the first method, as we have said before, has the advantage of operating at first with quite elementary matei-ial. Howbeit the common foundation of the two methods in our exposition appears to be first the theory of the ikosa- hedron, and then further the consideration of the rectilinear generators of the principal surface. That the first gives the actual normal equations to which we must once for all reduce the solution of the equation of the fifth degree, I cannot doubt. On the other hand, I form a different estimate of our geometrical reflexions and constructions, useful as they have been to us. I believe that we shall be enabled to develop the general theory 282 THE GENERAL EQUATION OF of form-problems algebraically, and in sucb wise that onr reduction of equations of the fifth degree to the ikosahedron appears as a mere corollary, and does not need to be established in a special manner. I have myself attempted this in Bd. xv. of the Mathematische Annalen, where I brought the connection between the problem of the A's and of the equation of the fifth degree — and, in fact, the formulae of Brioschi appertaining hereto as well as our formulae with the a^^'s — to the single fact that the substitutions of the A's can be co-ordinated to the even permutations of the x'b isomorphously and uniquely.* My reason for not entering upon these matters in the foregoing exposition is that I do not consider these wider speculations to which I have referred in I. 5 (§§ 4, 5, 9) as yet conclusive. I have the more readily confined myself to geometrical construc- tions of individual characteristics, believing that it is just by these that we shall be able to pass to a true insight into the general theory. § 9. On the Necessity of the Accessory Square Eoot. We are at the end of our exposition ; what we have yet to add concerns the necessity of that accessory square root which occurred in our first method in the Tschimhausian transforma- tion, in our second method, when we wished to effect the solu- tion of the problem of the A's. We shall show that this square root is in fact indispensable if an ikosahedral equation is to be reached at all ; we shall further prove that from this follows that theorem of Kronecker's which we have mentioned in II. 1, § 7, and which declares the general impossibility of a rational re- solvent with only one parameter for the ordinary equation of the fifth degree. In order now to prove the first point, let us formulate our assertion as follows. Let Xg, x^, . . . x^, be any five variable magnitudes, (p, ^fr, two integral functions thereof without a * "Ueber die Auflosung gewisser Gleichungen vom siebenten und achten Grade" (1879). See especially §§ 1-5. The mode of expression in the text supposes that to every permutation of the a:'s corresponds only one substitution iif the A's ; single-valuedness in the reciprocal sense occurs also, but it would not be necessary for the success of the algebraical process. THE FIFTH DEGREE. 283 common divisor. ITien it is impossible, we assert, to choose , i/r, in such wise that (29) ^^^JViMS) undergoes the ikosahedral substitutions for the even permutations of the x's. The proof presents itself at once if we consider that the original question, one belonging to the theory of functions, is transformed by virtue of the arbitrary choice of the x's into a question of the theory of forms. Namely, if, corresponding to any permutation of the sc's, the substitution formula : (30) x' = f, = 4±^ were to occur, we could at once, on account of the arbitrary nature of the x's, write (31) '^0{a4> + /3-^), 4' = C'(7<^ + 34), understanding by C an appropriate constant, so that, therefore, with the permutations of the x's, the two integral functions (j> and yjr are transformed hilinearly. But now, as we showed in detail in I. 2, § 8, every group of binary substitutions which is to be isomorphous with the group of non-homogeneous ikosa- hedral substitutions contains, of necessity, tnore than 60 opera- tions, while to the 60 even permutations of the x's not more than 60 transformations of the integral rational functions ^, i/r, can correspond. This is an insurmountable contradiction, and therefore the method proposed in (29) is, in fact, proved to be impossible, q.e.d.* The contradiction is not even removed if we now assume 5'x = 0, for every equation of the fifth degree can be transformed rationally into one with 5*35=0. For the sake of a better grasp of the essence of the proof, let us compare the theory of the principal equations of the fifth degree. In them we have, besides Xx = 0, Xx- = also ; let us therefore write equation (30) as follows : (32) 4>'{y + h^) = -\'{a4> + li-i^), * Cf. here and in the following paragraphs my oft-cited memoir in Bd. xii. of the Math. Annalen (1877), and also my communication to the Erlanger Societiit of January 15, 1877. 284 THE GENERAL EQUATION OF then, in the case of principal equations, it is by no means necessary that the two surfaces : 4>'{y4>+&-^) = Q, •4.' (a<^ + /34) = are identical with one another, but only that they intersect in the same curve the principal equation of fJie second degree repre- sented hy those conditions. Now we have in any case decided that ^, T^, and so likewise ^', i^', have no common divisor. We shall also require that no factor shall be capable of being detached when we modify the functions which arise from the addition of proper multiples of Xx, 'Sx?. Nevertheless, the curves of intersection of the principal surface with <^' = 0, i/r' = 0, may have a portion in common ; this portion must be only an incomplete curve of intersection, and must not admit of being traced out by a surface appended to the principal surface. If we assume that this is the case, no ground appears in fact for the existence of formula (31) (from which we deduced the contradiction). I must omit to work out in greater detail what I have said, and to show that in fact, in the reflexions thus given, our former treatment of the principal equations of the fifth degree is absorbed. The proof which we have given of our primary assertion is extended without any important modi- fication to other cases also. First, we might substitute at once the problem of the A's in place of the general equation of the fifth degree : we learn that it is impossible, in reducing this problem to an ikosahedral equation, to dispense with the square root si A (or an equivalent irrationality) as previously employed. We learn, further, that it is impossible to reduce the general equation of the fourth degree, by means of rational construction of resolvents, to an octahedral equation, or even, after adjunc- tion of the square root of the discriminant, to a tetrahedral equation.* Moreover, we can now make a practical application * As regards equations of the fourth degree, a solution can be effected in their case, as I here cursorily indicate, with the help of the octahedral equation (or of the tetrahedral equation), which is, so to say, a blending of the two methods, which for equations of the fifth degree are distinct. Denote as before the roots x„, xi, x-2, Xj, which are to be subject to the condition S.-<;=0, by quadrilateral co-ordinates in the plane. Then we have the principal conic 2a:^ = 0, and we saw above (II. 3, § 2) how a point belonging to it can be determined directly by an octahedral equation or a tetrahedral equation. We shall now co-ordinate to an arbitrary point x of the plane a point y of the principal conic as a covariant, by THE FIFTH DEGREE. 285 of our train of thouglit. In this respect I only remark that the property of A^, Aj, A,, which was described just now (§ 5), may be deduced in the way thus indicated. § 10. Special Equations of the Fifth Degree which can BE Rationally Reduced to an Ikosahedral Equation. We must now interrupt our general considerations, and make mention of special equations of the fifth degree which furnish an exception to the theorem just proved. In II. 2, § 4, we have given a geometrical interpretation of the resolvents of the fifth degree, and have seen that they can be represented by two half- regular twisted curves of deficiency zero. Our object now is to reverse this result. Let : (33) F{z,Z) = be an equation of the fifth degree with one parameter which admits of an interpretation of the kind mentioned. I assert that we can always reduce it by rational means to an ikosahedral equation. The proof is essentially the same as we have given in a some- what different form in II. 3, § 1, in considering the principal equation. By hypothesis the five roots of (33) admit of repre- sentation as rational functions of an auxiliary magnitude X : (34) x, = RjK), in such wise, that for appropriate variation of \ the a;^'s under- go any even permutation. We must now apply the proposition from the theory of rational curves, that this \ can always be drawing from x the two possible tangents to the conic, and choosing one of the two points of contact. We can then establish the octahedral equation (or tetra- bedral equation) on which y depends, and hence by inversion find x, &c., &c., all in strict analogy with the developments which we have opened up in the two concluding paragraphs of the preceding chapter. In the case of equations of the third degree, all such prolixity, as we remarked in II. 3, § 2, disappears. In fact, we saw in I. 2, § 8, that the dihedral group of six substitutions, which comes under consideration in connection with them, can be very well transformed into the homogeneous form, without the number of its substitutions being increased ; the grounds for the occurrence of the accessory irrationality which we have recognised as appropriate for equations of the fourth and fifth degrees are tlierefi,re wanting. 286 THE GENERAL EQUATION OF introduced as a rational function of the x's, and therefore in such wise that to every point of the curve corresponds only one \.* I will assume, for the sake of brevity, that the X appearing in (33) is already chosen in the manner here indicated. Then every one-valued transformation which transforms our curve into itself, in particular therefore every even permutation of the xja, establishes for X, a one-valued transformation having a one- valued reciprocal and therefore a linear transformation. Thus we obtain corresponding to the 60 even permutations of the xja a group of linear substitutions, holohedrically isomorphous with them, of the variable \. By I. 5, § 2, this is of necessity the ikosahedral group ; it appears in the canonical form which we have always maintained as soon as we introduce in place of X a proper linear function X' = ^^ — -= as parameter. ITiis X', which is itself a rational function of the x^'s, then depends directly on an ikosahedral eqiiation, whereupon the proof of our assertion is disposed of. We append to what has been said a few stray remarks. First we see that we can reiterate our theorem with unimportant modifications in the problem of the A's, or, if we like to take into consideration the octahedron or tetrahedron instead of the ikosahedron, in the equation of the fourth degree. We recognise, further, that for the equation of the fifth degree there can be no rational twisted curves .which for the whole of the permutations of the xjs pass over into themselves. Finally, we remark that the occurrence of rational invariant curves (as we will express it) is altogether limited to those form-problems of which the group is holohedrically isomorphous one of the group of linear substitutions of a variable which we have pre- viously enumerated. § 11. Kronecker's Theorem. We have now all the requisite materials for completing the proof of the ofb-mentioned theorem . of Kronecker. Our object is to prove that it is impossible, in the case of any proposed equa- * Cf. the proof of this theorem in Luroth's paper in Bd. ix. of the Mathema- tische Annalen (1875). THE FIFTH DEGREE. 287 tion of the fifth degree, even after adjunction of the square root of the discriminant, to construct a rational resolvent which contains only one parameter. Let us first remark that we can at once impart an apparently more precise formulation to this theorem, inasmuch as the group of the even permutations of five things is primitive.* Namely, we shall be able to derive, on the grounds stated, from every rational resolvent a fresh equation of the fifth degree F{X) = Q by means of renewed resolvent construction ; and here we may at once subject the JT's to the condition 'SX=0. The roots X„ are here severally co-ordinated to the original x„'s in such wise that the co-ordination remains unaltered for any even permuta- tions of the xj&. We can therefore write as before : (35) X =p . x™ + q ■ a;|? + r ■ zf + s ■ x't\ where a;* = a^— -5*0:^ and p, q, r, s, depend rationally on the coeflScients of the proposed equation and the square root of the corresponding discriminant. All that we now have to shoiv is this : that it is impossible to form from, the general equation of the fifth degree, hy means of a Tschirnkausian transformation (35), an equation of the fifth degree with only one parameter. To this end we first reflect generally as to what geometrical interpretation such an equation would have to receive. The totality of the arbitrary values x,,, Xj, x^, Xg, x^, form a simply con- nected continuum. If we therefore allow x^, x-y, . . . x^, in (35) to alter arbitrarily, the point X will, at all events, trace out an irreducible locus. If we now add the supposition that the equa- tion of the XJs contains only one parameter, the irreducible locus in question will have to be a curve. I say now that the irreducible curve so obtained ivill be transformed into itself for the 60 even collineations of space. In fact, in virtue of the con- vention which we have made concerning the coefficients p, q, r, s, occurring in (35), the even permutations of the X^'s correspond to the even permutations of the xja ; while, on the other hand, we can attain to every permutation of the xjs (and therefore in particular to every even permutation thereof) by allowing the * Cf. the definition in I. 1, § 2. 288 THE GENERAL EQUATION OF xja, beginning from any initial values, to move continuously in a suitable manner. We now return specially to the developments of tlie preceding chapter. Namely, it is evident that the curve of the XJs just described must in every case he rational. For we can suppose the x^, x^, . . . x^, in (35) rationally dependent in some way on a parameter X, whereupon the X^'s themselves become rational functions of this \ : we need not regard the objection that in special cases the \ may altogether disappear from the JlJs, since we can evidently always avoid such a contingency. The premises of the preceding paragraph are therefore in fact given. We conclude that we can establish a rational function of the XJs ivhich for the even 'permutations of the XJs undergoes the ikosahedral substitutions. This function would by virtue of (35) also depend rationally on the xjs. in such wise that it would undergo ikosahedral substitutions for the even permuta- tions of the xjs. But now we have expressly proved in § 9 that such a rational function of the xja is impossible. We therefore arrive at a complete contradiction, and must therefore give up our assumption that a Tschirnhausian transformation (35) exists with the property more precisely described above, q.e.d. I conclude by adding a few more general observations on the theory of equations. First, if in the foregoing exposition we substitute through- out the octahedron or tetrahedron for the ikosahedron, we can repeat all our considerations unaltered for the equation of the fourth degree till we come to the one that treats of the primitivity of the corresponding group. The group of the equation of the fourth degree is composite. If, therefore, we wish to recover Kronecker's theorem for the equation of the fourth degree, we must expressly attach to them the condition that the group of the resolvent coming under consideration is to be holohedrically iso7norphous ivith the group of the twenty-four or the twelve per- mutations of Xj, Xj, X.2, Xj, x^. If we leave out this condition, rational resolvents of the general equation of the fourth degree may very well occur which contain only one parameter. The empirical proof thereof is effected by the ordinary solution of the equation of the fourth degree. In fact, this operates merely THE FIFTH DEGREE. 289 with auxiliary equations which contain only one parameter, namely, with binomial equations. In the case of equations of the third degree, there can, of course, on the grounds of our previous remarks, be no question of a theorem corresponding to Kronecker's. Concerning equations of a higher degree, I will here, in order not to be prolix, only make one remark, retaining therein, for the sake of simplification, the restriction which we formulated just now for the fourth degree. On the supposition mentioned, resolvents with only one parameter — disregarding special and easily-recognised cases — are impossible in the case of the general equation, for the reason that, according to the observation of § 10, among the corresponding invariant curves no rational ones can exist. appe:n^dix. BY THE TRANSLATOR. Note A. — Glass models of the Ikosahedron are sold as letter- weights by stationers for eighteen pence. They may be ob- tained at No. 355 Strand. Note B. — The formulae of I. 1, § 3, are illustrated by the following problem, which appeared in the Educational Times : — Take three points on a sphere : _ _ (f? + ic) i'l - (b — ia) _ ((V + ic) 2., - {V - ia) 1' ^i~ {1 + ia) z^ + d-ic ' ^~ {b' + ia')z.-^ - {cV - ic) _ {d" + ic") 2j - (h" - ia") (b" + ia") z^ — (d" - ic")' and show that the polar triangle is formed by the points : (i + iy)Z,-{l3-ia ) 1' 2 (fi + ia)Z^ + {&-iy}' ^ (&' + iy')Z, -i/3'-ic. ') ^ {&" + iy") Z,-(fi"- ia") ■^ (/3' + ia) Z, + (3' - iy') iP" + ia") Z^ + (&" - iy')' where a" ;8" y" 3" bc'-b'c ca'-c'a ab'-a'b J (I - d^j {1 - d^-) - aa - bb' - cc' Note C. — Mr. Greenhill points out that some interesting numerical cases are obtained for the resolvent of the r's, I. 4, § 10, by putting: r = 3 or 11 or 19. Note D. — I collect here such data as I can obtain wi;h regard to the important advance which has recently been maiie 292 APPENDIX. by Professor Klein and his pupils in a most interesting field of analysis. The connection of the twenty-seven straight lines on a cubic surface with the trisection of the hyperelliptic functions of the first order has long been known ; it is treated from the point of view of the theory of substitutions in the " Traits des Substitutions " of M. Camille Jordan. In a recent article in the Jourjial de Mathematiquts (4° s^rie, tome iv., fasc. ii., 1888), " Sur la resolution, par les fonctions hyperelliptiqnes, de I'^qua- tion du vingt-septieme degre, de laqnelle depend la determina- tion des vingt-sept droites d'nne surface cubique," Professor Klein indicates briefly how the methods employed in this treatise may be extended to the equation of the 27th degree. The details will shortly appear in the Mathematische Annalen. Meanwhile reference should be made to the Inaugural Disser- tation of Dr. Alexander Witting (Teubner, Dresden), " Ueber eine der Hesse'schen Configuration der ebenen Curve dritter Ordnung analoge Configuration in Baume auf welche die Transformationstheorie der Hyperelliptischen Punctionen {p = 2) fiihrt." To these sources I am indebted for much of what follows. We consider the integrals where f(z) is a rational integral function of z of the sixth degree. Between the four pairs of periods of these integrals Wjo to^t, (i=l, 2, 3, 4) there exists the relation : If we take a new pair of integrals ?^,'-22 - "o'-is _ - Mi^Oji + ItjWii V. — > ^2 '-ll"22-'^12*'21 > '-n»'22 - '"l2»'21 periods of v are : 1 n . '"l3'^22-'-12'"23 . _ '^14'«'22 - '>'12"24 '^^ '»'ll'»'22 - "12< 0, 1, .„._"""" -"^^"^ -' '*'ll'"22-»'l2"2l 22 a<„a,,2-!.>,2<«.,j' APPENDIX. 293 and the identical relation is ''12 ~ ''2r A double 3-function of the first order with characteristic pi' 92\ is ^fJv 92\ ("k "2' ''ll' '^12' '22) + » : eT(»l''l+ff=''=)2'«l'"!( - l)''l''>+"!*' e'^C^"! +?!)''. +(2''2+17j)'-j) where Here each of the letters in the characteristic may have the value or 1 ; there are thus sixteen double 3-functions, ten even and six uneven, viz., an even function has the determinant ■'1' y-2 even, and vice versa. To each of the six uneven functions K ^2 corresponds a root of the sextic function, and to each of the ten even functions corresponds a grouping of the roots into two triplets. We now consider the four functions a, /3 = 0, 1; 1,0; 1, 1 ; 1, 2. p is an indeterminate factor, 3 any uneven 3-function. The quotients of these four functions undergo linear substitutions with constant coefficients for every linear transformation of the Tj^fe's which leaves the characteristic of the 3-function unaltered. It is shown in the " Traits des Substitutions " that the group of the equation of the twenty-seven straight lines on a cubic surface, after adjunction of a square root, is holohedrically isomorphous with the group of the 25920 fractional substitu- tions of the quotients of the z's. This latter group not being holohedrically isomorphous with the group of homogeneous transformations of the 2's, it is necessary to find one which is. To this end Professor Klein considers the six quantities formed from two sets of cogredient ;:''s. 294 APPENDIX. We have then a corresponding group of transformations on the co-ordinates a^^ of the linear complex The next step depends on a fundamental theorem given in Bd. XV. of the Mathematische Annalen. If to a group of sub- stitutions among n letters Xj . . . x^ there be co-ordinated in holohedric or merihedric isomorphism a group of linear trans- formations of /i letters 2/j . . . y^, it is always possible to find yu, functions of the xb, Pj . . . P"^, which, when the substitu- tions of the first group are effected on the x'a, are themselves transformed by the corresponding linear transformations of the 2/'s. Let us then form six functions of the roots x-^ . . . x^^ of the equation of the twenty-seven right lines, such that they undergo the transformations of the Uii^s, when the substitutions of the group of the equation are effected on the a;'s. These six functions being taken as the co-ordinates of a linear complex a, let us co-ordinate to this in a covariant and rational manner three other linear complexes a', a", a'", and then find two linear combinations of these. A, A', fulfilling the conditions 2A; = 0, 2AA;. = 0, 2A,= = 0, The first and third equations imply that they are to be special complexes, i.e., right lines, and the second equation that these lines intersect. Their point of intersection is the point z which we required, namely, it is covariant to the complex a. . We finally substitute for the co-ordinates z^, z^, z^, z^, their values as 3-functiona. n^lntkd hv haixantvne, hanson and co. kuinbi:kCjH ani> lonuon.