64,2/ ^suEauncs ^^^ ■. ^n^^Z >^W* y Date Due FEB 1 1 tPR.^ fc-r:. i ' ' '^m ^Wr- HOT Fe^— H9?"7 I1AR2 2l983f fi^^¥ SSf W n^=^' •Ml'jl l"||' '"'f IISl' 1987^ 1 DA3 A«^v« PRINTED IN (t?y NO. 23233 Cornell University Library QA 191.S42 A treatise on the theory of determinants 3 1924 001 569 148 Cornell University Library The original of tiiis book is in tine Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924001569148 A TKEATISE THEOKY OF DETEEMINANTS. CambtiSge: PRINTED BY C. J. CLAY, M.A., AT THE UNIVEKSITY PKESS. A TEEATISE ,,(„-fi-'-.,'t ■? 'ill ON THE THEORY OF DETERMINANTS AND THEIR APPLICATIONS IN ANALYSIS AND GEOMETRY, EGBERT FORSYTH SCOTT, M.A. OF Lincoln's inn ; FELLOW OF ST JOHN'S COLLF.GE, CAMBRIDGE. CamlirtUgE : AT THE UNIVERSITY PRESS. aotiDon: CAMBRIDGE WAREHOUSE, 17, PATERNOSTER ROW. ffiambriUse: DEIGHTON, BELL, AND CO. JUipjIa: F. A. BEOCKHAUS. 1880 T [All Rights reserved.] /V\,^2> PREFACE. In the present treatise I have attempted to give an exposition of the Theory of Determinants and their more important appli- cations. In every case where it was possible I have consulted the original works and memoirs on the subject ; a list of those I have been able to see is appended as it may be useful to others pursuing the same line of study. At one time I hoped to make this list exhaustive, supplementing my own researches from the literary notices in foreign mathematical journals, but even with this aid I found that it would be necessarily incomplete. In consequence of this the list has been restricted to those memoirs which I have seen, the leading results of which are incorporated either in the body of the text or in the examples. The principal novelty of the treatise lies in the systematic use of Grassmann's alternate units, by means of which the study of determinants is, I believe, much simplified. I have to thank my friend Mr Jas. Barnard, M.A. of St John's College and Mathematical Master at the Proprietary School, Blackheath, for the care he has bestowed on correcting the proofs and for many valuable suggestions. R. F. SCOTT. Feb. 1880. S. U. b CONTENTS. CHAPTER I. INTRODUCTION. ABT. PAOK 1 — i. Notations 1 5, 6. Permutations of Elements 3 7, 8. Effect of a simple Inversion 4 9 — 11. Cyclical permutations .'" . . 6 12. Definition of a Determinant 7 13. Notations for a Determinant . 9 16 — 18. Alternate Numbers 11 19. Expression for a Determinant as a product of alternate numbers . 13 20, 21. Examples . . 14 CHAPTER II. GENERAL PROPERTIES OF DETERMINANTS. 1, 2. Interchange of rows and columns . . . . 17 8, 4. Value of a Determinant when the elements of a row are sums . 18 5, 6. Examples . .19 7. Solution of a system of linear equations 21 CHAPTER III. ON THE MINORS AND ON THE EXPANSION OF A DETERMINANT. 1. Number of Minors of order jj . . 24 2 — 4. Complementary Minors . .... . . 24 CONTENTS. ABT. PAGE 5. Laplace's Theorem . . . 26 6, 7. Examples .... ... . .28 9, 10. Expansion of a Determinant aeoorcling to the elements of a row . 30 12—14. Examples . • • 32 15, 16. Differential Coefficients of Determinants . . . . 36 18 — 21. Albeggiani's expansion of a Determinant with polynomial elements . 38 22, 28. Expansion of a Determinant according to products of elements in the leading diagonal . • 40 24. Cauchy's theorem ■ • 43 25. Examples ... ... . 43 CHAPTER IV. MULTIPLICATION OP DETERMINANTS. 1 — 4. Determinant formed by compounding two arrays . . 46 5—7. Examples .... ... . 47 8. ■ Fundamental theorem deduced from Laplace's theorem . . 51 9. Minor of a product Determinant . .... 63 10. Differential Coefficient of a product Determinant . . 54 CHAPTER V. DETERMINANTS OF COMPOUND SYSTEMS. 1 — 4. Eeoiprocal Ai-rays ... ... . . 55 5, 6. Eeciprocal Arrays of the first order . . .57 7, 8. Examples . . 59 9, 10. Eeciprocal Arrays of the m'^ order . 60 11 — 20. Theorems of Sylvester and Picquet ... 61 CHAPTER ri. DETERMINANTS OF SPECIAL FORMS. 1 — 3. Symmetrical Determinants 67 4 — 8. Skew and skew symmetrical Determinants . . . 69 9 — 16. Skew symmetrical Determinants, Pfaf&ans . 71 17, 18. Skew Determinants, Examples .... ... 76 20 — 22. Orthosymmetrical Determinants 78 23 — 26. Determinants whose elements are arranged cyclically 81 27 — 30. Determinants whose elements are binomial Coefficients . . 83 ABT. 1,2. 3. 4-9. 10—18. 19, 20. CONTENTS. IX CHAPTER VII. CUBIC DETERMINANTS. PAGE Definition, Notation .... 89 Expression for a Cubic Determinant as a product of alternate numbers ...... . . 90 Elementary properties of Cubic Determinants ... .90 Determinants with multiple suffixes . .93 Examples . . • 97 APPLICATIONS. CHAPTER VIII. THEORY OF EQUATIONS AND ELIMINATIONS. 1 — 6. Linear Equations ... . . . 99 7. Solution of a system of linear congruences 102 9 — 11. Eesultant of two equations . . . 103 12. Discriminant of a function . . . 106 13 — 15. Examples of Elimination . . .... 107 16, 17. Reality of the roots of the equation for the secular inequalities . Ill 18. Eurstenau's method of approximating to the least roots . . . 112 CHAPTER IX. RATIONAL FUNCTIONAL DETERMINANTS. 1—3. Product of all the differences of n numbers 115 4—17. Examples of functional Determinants 116 CHAPTER X. JACOBIANS AND HESSIANS. 1. Definition and Notation for a Jacobian . .... 129 2. The Jacobian of dependent functions vanishes . . .129 CONTENTS. ABT. PAGE 3, 4. Jacobian of functions with a common factor . • ■ 130 6 — 10. Properties of Jacobians .... ■ • 131 11. Bertrand's definition • ^36 12. Definition by means of alternate numbers 137 13. 14. Transformation of a multiple Integral . • • 138 15. Definition of a Hessian, Example . . • 1^2 16. Jacobians and Hessians are Co-variants .... . 143 17. Jacobian of n linear functions, Hessian of a quadrio . . .145 CHAPTER XI. THEORY OF QUADRICS. 1 — 3. Eeciprocal Quadrics . . 146 4 — 8. Eesolution of a Quadrio into the sum of squares. Darboux's method 147 9, 10. Bepresentation of two quadrics as the sums of squares of the same hnear functions 152 11—17. The Orthogonal Transformation, Examples . . . 154 18. Aronhold's invariant of three ternary Quadrics . ... 159 CHAPTER XII. DETERMINANT OF FUNCTIONS OF I'HE SAME VARIABLE. 1 — 9. Definition, Elementary properties . 160 10. Application to linear differential equations . . 166 11. Hesse's Solution of Jacobi's equation . . . 167 CHAPTER XIII. CONTINUED FRACTIONS. I. Ascending and descending Continued Fractions .... 169 2 — 7. Expression for the Convergents to descending Continued Fractions, Elementary properties 170 8, 9. Ascending Continued Fractions, Transformation to descending Con- tinued Fractions ... 174 II. Fiirstenau'g extended Continued Fractions . ... 177 CONTENTS. CHAPTER XIV. APPLICATIONS TO GEOMETRY. ART. 1 — 3. Area of a triangle, Volume of a Tetrahedron 4 — 6. Elementary Identities . ... 7. Application of Alternate Numbers 8 — 14. Angles between straight lines, Solid angles, Spherical figures . 15 — 18. Systems of straight lines, Co-ordinates of a line, Mutual momenta 19 — 23. Belation between the lines joining five points in space (Cayley) Volumes of tetrahedra. Areas of triangles, Siebeek's theorem 24 — 28. Formulfe relating to an Ellipsoid, Theorem of Cayley for six points on a Sphere, Paure's theory of Indices .... 29—39. Systems of spheres, Mutual powers, Common tangents 180 183 185 187 193 196 206 Examples .... . . .... . 213 BiBLIOOBAPHT ... 242 THEOKY OF DETEEMINANTS. CHAPTER I. Introduction. 1. The object of the theory of Determinants is to obtain compendious and simple methods of dealing with large numbers of quantities. In the words of Professor Sylvester, "It is an algebra upon an algebra ; a calculus which enables us to combine and foretell the results of algebraical operations in the same way as algebra itself enables us to dispense with the performance of the special operations of arithmetic." It will be found that the advantages and success of the method depend in great measure upon the notations which have been employed. 2. To indicate concisely the quantities discussed different notations have been used. The numbers belonging to the same class being denoted by the same letter, the different numbers of that class are distinguished by affixing numbers or letters, e.g. (t'V ^2' «3' ••■ ^n' denotes such a class of numbers. Each letter with its affix is called an element ; the affixed number from its position is usually called the suffix of the element. S. D. 1 2 THEORY OF DETERMINANTS. [CHAP. I. We have frequently to deal with a series of such classes, each containing the same number of elements; these when written one under the other in rows form a rectangular array, the class being denoted by the letter while the suffix indicates the position of the element in the class. E.g. «!, Ctaj «3 iv K, K Ci, Cj, C3 3. In the theory of determinants we have frequently to deal with several such arrays, and it will be found that the most con- venient notation is the following : *21' ^i2' ^2S' ••• ^2J" where there are m horizontal and p vertical rows of el«ments. Then a^ is that element in the array of a's which is situated at the intersection of the k"" horizontal and s^ vertical rows. The first suffix tells us the horizontal and the second suffix the vertical row in which the element stands. In the present work these horizontal and vertical rows will be called rows and columns ; a^ therefore stands in the k^ row and s*^ column. Occasionally when we are dealing with a single array the letter is omitted, and instead of a^ we write (ks) only. Such a notation is called an umbral notation, (ks) beiag not a quantity, but, as it were, the shadow of one. 4. To give an example of the use of this notation take two groups of points in space, the first consisting of m and the second of p points. Then we may denote the distance between the A"" point of the first group and the s*" point of the second by d^, or , 2 — 6.] PERMUTATIONS OF ELEMENTS, 3 (ks) simply, and the whole set of lines joining the points of the two groups would be denoted by the array in § 3. At the same time the meaning of any selected element d^^ is perceived at once. 5. If we have any n elements a^, a^, ... a„, we may call where the elements are arranged according to the magnitude of the numbers forming the suffixes, the natural or original order of the letters. Any other order is called a permutation of the elements. One element is said to be higher than another when it has the greater suffix. When in any permutation an element jvith a higher suffix precedes another with a lower we have an inversion. Thus the permutation a^, a^, a^, a^, of four letters, contains the following four inversions, «4"2' «A. «A' "'2^'V where we compare each element with all that follow it. Following Cramer it is usual to divide the permutations of a given set' of elements into two classes ; the first class contains those permutations which have an even number of inversions, the second those which have an odd number. 6. By permutating the elements a^, a^,...a^ we obtain all possible ways in which they can be written. The same result is arrived at by writing down all the permutations of the suffixes 1, 2, ... w and then putting a's above them. By repeated interchange of two suffixes we can get every permutation of the given elements from their original order. For if we start with two suffixes 1, 2, they have but two arrangements, 1, 2, 2, 1, of which the second is got from the first by a simple interchange. Taking three elements 1, 2, 3 out of these we can select the duad 2, 3, whose permutations are 2, 3 ; 3, 2. Prefixing 1 to each of these we get 1, 2, 3; 1, 3, 2, which are two permutations of the 1—2 4 THEORY OF DETERMINANTS. [CHAP. I. given elements. Proceeding in like manner with the other duads 1, 3; 1, 2, we get the six arrangements of three figures 12 3, 13 2, 2 3 1 2 13, 3 12, 3 2 1. Next take four numbers 1, 2, 3, 4. We get four triplets by leaving out one number, viz. 12 3, 12 4, 13 4, 2 3 4. For each triplet we can write down six arrangements by the rule just given for three numbers, then adding on the missing number we get twenty-four arrangements of four numbers, viz. 12 3 4 12 4 3 13 4 2 2 3 4 1 2 13 4 2 14 3 3 14 2 3 2 4 1 13 2 4 14 2 3 14 3 2 2 4 3 1 3 12 4 4 12 3 4 13 2 4 2 3 1 2 3 14 2 4 13 3 4 12 3 4 2 1 3 2 14 4 2 13 4 3 12 4 3 2 1. And so we could go on to write down the arrangements of any set of elements. The number of arrangements of n letters is 1 . 2. 3...n or n! V an even number. ■>i' 7. If in a given permutation two elements be interchanged while all the others remain unaltered in position, the two resulting permutations belong to different classes. This will be proved if we can shew that the difference between the number of inversions in the two permutations is an odd number. We can represent any permutation of a group of elements by A d B e C (1), where d and e are the two elements to be presently interchanged, A the group of elements which precede d, B the group between d and e, and G the group which follows e. The permutation we obtain is A e B d C (2). 6 — 8.J PERMTTTATIONS OF ELEMENTS. 5 The number of inversions in the two permutations (1) and (2) due to the elements contained in the groups A, B and G is in each case the same. And since the elements of A precede d and e in both permutations we get no new inversions in (2) from these ; the elements of G follow both d and e, and therefore give rise to no new inversions. We have therefore only to consider the changes in the two permutations d B e and e B d (3). Suppose that e is higher than d; let -B contain I elements of which 6j are higher than d and l^ higher than e. Then in the permutation d B e -we have, independently of the inversions con- tained in B itself, h — h^ + b^ inversions, because there are h — h^ elements lower than d and h^ higher than e. In eB d we have b—b^ inversions on account of e, b^ on account of d, and one because e is higher than d; thus, without counting the inversions in B, we have b — b^+ b^+ 1. The difference between the number of inversions in the permutations (3), and therefore in (1) and (2), is thus 5-6, + 6, + l-(&-5, + 6,) =2(6, -&,) + !, which is an odd number, shewing that the permutations belong to different classes. / 8. The same result may be arrived at as follows. If there be n quantities whose natural order is and if in any arrangement we subtract each suffix from all that follow it and multiply these differences together, we shall have a product whose sign will depend on the number of inversions in the given arrangement, the sign being positive if the number of inversions is even and negative if the number of inversions is odd. If then i, k be any two suffixes chosen arbitrarily which are to be interchanged, i preceding i in the given arrangement, the product of the differences will consist of four parts. (i) The factor k — i. (ii) and (iii) A set of factors such as r — k, and r — i, where r is some number of the series \...n excluding i and k. t. YA 6 THEORY OF DKTERMirA.NTS. [CHAP. I. (iv) A set of factors such as r - s, where r, s are any two numbers of the series 1, 2...w excluding i and k. Then for the given arrangement the product of the differences will he ±{k-i)JI{r-€)[r -k)ll{r - s), where the symbol 11 stands for "the product of all such factors." If now we interchange i and k, the signs of aU factors such as (r -k){r- i), {r - s) remain unchanged, while k-i changes sign. Thus on interchanging two elements the product of the differ- ences changes sign, i.e. by interchanging two suffixes we have introduced an odd number of negative factors and therefore of inversions, hence the two arrangements considered belong to dif- ferent classes. 9. If in a series of elements each is replaced by the one which follows it, and the last by the first, we are said to have got a cyclicalj)ermutation of the given arrangement. If the system of elements "^ be considered as forming an endless band, if we cut this band between a, and a^ we have the natural order, cutting it between Kj and ftj we have a cyclical permutation of the first order, and so on. Such a cyclical permutation is equivalent to n — 1 simple interchanges, viz. we move a^ from the first to the last place by interchanging the first and second elements, then the second and third, and so on, in all n — 1 simple interchanges. Thus a cyclical permutation of a given arrangement belongs to the same or opposite class as the given one according as the number of ele- ments is odd or even. 10. Every permutation of a given set of elements may be considered as derived from a fixed permutation by means of cyclical permutations of groups of the elements. This is best illustrated by an example. Let the suffixes of two permutations of nine elements be 7, 6, 3, 2, 1, 4, 8, 5, 9 8, 7, 9, 5, 1, 6, 4, 3, 2 / 8 — 12.]' DEFINITION OF A DETERMINANT. 7 Here the second permutation is obtained by replacing in the first 7 by 8, 8 by 4, 4 by 6 and 6 by 7, which completes a cycle. Then 3 is replaced by 9, 9 by 2, 2 by 5 and 5 by 3, which completes the second cycle. Lastly, 1 forms a cycle by itself. 11. If elements which remain unchanged like 1 in the preceding example be considered as forming a cycle of one letter, we may state the following theorem: Two permutations belong" to the same or different classes, according as the difference be- tween the number of elements and the number of groups by whose cyclical interchange one permutation is got from the other, is even or odd. For if there be n elements altogether, and p cycles of n^, n^.-.Tij, letters, the cyclical interchanges are equivalent to {n^-l)+{n.^-l) + ...-V{n^-\)=n^ + n^...+n^-p = n—p simple interchanges, which proves the theorem. In the example in Art. 10, w = 9, p = 3, and thus they belong to the same class. 12. If the number of rows and columns in an array be the same, we have a square array. Let such an array, containing n' elements, be ^^21' "^22 ""in' The diagonal of elements a^^, a.^ ... a„„ will be called the leading or principal diagonal. _4_j;sj±aiiu-iaaet ipn, which i s called a de terminant^ can be formed with the elements of this array as follows : From the array choose n different elements such that there is one and only one element from each row and column, multiply these elements to- gether, the product will be a term of the determinant of n letters. For example, the set of elements Oi,,t Oi„. tt„„, 8 THEORY OF DETERMINANTS. [CHAP. I. situated in the principal diagonal of the square array, form a term of the determinant ; this will be called the leading term, and to it we assign the positive sign. The sign of any other term is determined as follows. From the mode in which the elements were selected, it follows that /, h ...s, and g, k ... t are each of them permutations of 1, 2 ...w. Let them contain p and q inversions respectively, then the sign of the term is (— 1)^'. The sum of all the possible terms with their proper signs is the determinant of the array. More simple rules may be given for determining the sign of any term. If we interchange any two elements a^ and a,,- the term does not change its sign. For this interchange is equivalent to the interchange of i with h and j with k. By these two in- terchanges we increase both ji and q by an odd number, and hence the sign of the term is unaltered. It is therefore usual to give to one series of suffixes their natural order, when one of the two numbers ^ or g' is zero, and the sign of the term of the deter- minant depends solely on the number of inversions in the other series, and is the same whether the first or second series of suflSxes retains its natural order. It is thus clear that all the terms of the determinant will be obtained from the leading term by keeping the first suffixes fixed in their natural order, and writing for the second suffixes in succession all possible permuta- tions of the elements 1, 2 ... w, giving to the product of the elements the positive or negative sign according as the number of inversions is even or odd. Such a determinant is said to be of the w*'' degree, since each term is the product of n elements. It has nl terms in all, since this is the number of permutations of the second suffixes, each 12—14.] NOTATIONS FOR A DETERMINANT. of whicli gives a term of the determinant. One half of these terms have the positive, the other half the negative sign. 13. Various notations are employed for the determinant of a system of r^ elements. Cauchy and Jacobi denoted it by drawing two vertical lines at the sides of the array, or by writing + before the leading term and prefixing a summation sign, «2,. «., a,, = S + aiia,j...a„ Sylvester uses the umbral notation 1, 2, 1, 2 n, n. If the determinant be written in the form a'u 2/.> ^1 y^: ^), we may denote it by I ^'i, Vi, ^i ■■■ I (»■ = !, 2 meaning by this that i is to take the different values 1, 2 ... n in succession. Lastly, the determinant with double suffixes may be denoted by I ctft 1 ii ^=1. 2 ... n), the bracket at the side telling us what values the suffixes i and k take. This bracket is frequently omitted in practice. This notation is, I believe, due to Prof. H. J. S. Smith, who employs it in his report on the theory of numbers, Brit. Ass. Rep., 1861, p. 504. 14. From Art. 6 we know the permutations of a system of two, three, or four elements. These give us the determinants of degree two, three, and four, viz. 10 THEORY OF DETEEMINANTS. [chap. I. "-11' "-li 11' «12. «13 '21' a^, «23 SI' 0^32' 0^33 «1' 6.' Cl. ^1 «2' &.' ^2' ^. «3' h> ^3' ^3 «4' K C4' ^^4 = «11%«33 - '^12C^21«33 +, «1S«21«32 - <^n«^23«32 + «12«23«81 " «13«22^31 ' ; ' I, 1/ - ^ + '^h^^A - (^sK^'A - <^i^2cA + (^J>^Ai, + "'A'^A ^ afifA - (^fifA + o-PfA + «i^3C4^2 - «s^icA - (ifi^pA+^h'^A + "a^Af^a - «ACi^2 - (^h'^A +^K<^A + «^2V3<^1 - «4^2C3^1 - ^^3^2^! + <^h^A- A useful mnemonical rule for writing down the expansion of any determinant of the third order is the following, due to Sarrus. Let the determinant be «2. K ^2 Alongside of this repeat the first and second columns in order «i \ c, a. \ \ X X y «2 ^2 ^2 "-2 K' / X X N «3 ^3 C3 «3 &3 and form the product of each set of three elements lying in lines parallel to the diagonals of the original square. Those which lie in lines descending from left to right have the positive, the others the negative sign. Thus the determinant is «Ac3 + ^/2«3 + CiaA ■cX%-<^i^h-\<^i<^s- In practice it is not necessary actually to repeat the columnsi but only to imagine them repeated. 14—16.] ALTERNATE NUMBERS. 11 It is not difficult to devise similar rules for determinants of higher order than the third, but we shall obtain methods for reducing the expansion of a determinant to that of several deter- minants of lower order, and for reducing the order of a determi- nant, so that they are unnecessary. 15. If we interchange rows and columns in the determinant of Art. 13, we get This is the same as the original determinant with the suffixes of each element interchanged. Its expansion is then obtained from that of the original determinant by interchanging in each term the suffixes of each element. That is to say, in the term ttjj, a^j ... a„„ we keep the second suffixes fixed in their natural order and write for the first suffixes all possible permutations of 1, 2 ... n. But the reasoning of Art. 12 shews that each term in the new determinant has the same sign as the corresponding one in the original determinant. Thus a determinant remains unchanged in value when its rows and columns are interchanged. Alternate Nwnbers. 16. The magnitudes with which we deal in ordinary or arithmetical algebra are subject, as regards their addition and multiplication, to the following principal laws : (i) The associative law, which states that {a + l) + c = a + (b + c) = a + h + c, or that ab.c=a. bc = abc. (ii) The commutative law, which states that a + h = b + a, ab = ba. 12 THEORY OF DETEEMINANTS. [CHAP. I. (iii) The distributive law, which states that {b + c)a = ba + ca, a{b + c) = ab + ac. The researches of modern algebraists have led them to con- sider quantities for which one or more of these laws ceases to hold, or for which one or more of these laws assumes a different form. Numbers, whether real or ideal, which follow the laws of arithmetical algebra will be called scalar quantities. We shall find it useful to consider a class of numbers which have received the name of alternate numbers. These are deter- mined by means of a system of independent units given in sets like the co-ordinates of a point in space; such a set will be denoted by e^, e^, ... e„. A number such as formed by adding the units together, each multiplied by a scalar, will be called an alternate number of the «.* order. In combination with scalar quantities and with units of other sets these units follow the laws of ordinary algebra. In combina- tion with each other the units of a system follow the associative law and the commutative law as regards addition, but for multi- plication we have the new equation efi, = -eA (1). As a consequence of which it follows at once that e.' = (2) for all values of i. 17. If A = a^e^ + a^e^+...+ a^e,^ , £ = \e^ + b,e^+... + b^e^ be two alternate numbers of the n'" order, we define their product as follows : AB = 'Za^e^tb^ej i i = Sajfij . bfii = lapfite^. 16 — 19.] ALTERNATE NUMBERS. 13 Hence, by equations (1) and (2) of Art. 16, AB = (afi^ - a,6 J e^e^ + {aj>^ - a^) e^e^ + ... Thus clearly AB = -BA and A'' = 0, proving that alternate numbers have the same commutative law of multiplication as the units. This kind of multiplication, where AB = - BA, is called polar because the product AB has opposite properties at its two ends. 18. If & be any scalar {A +kB)B = AB + JcB' = AB, so that the product of two alternate numbers is not altered if one be increased by a multiple of the other. If we have a product of more than two numbers ABC L, it follows that for one of them, say C, we can write C+Jc^A+ k^B + . . . + k,L, and the product will still remain unaltered. The alternate numbers belong to that class of algebraical magnitudes for which multiplication is a determinate, but division an indeterminate process. Viz. -g- = A+kB, where k is an arbitrary scalar. The continued product e^e^ ... e„ of all the units of a set will in future be assumed to be unity. An explanation of this assumption will be given later on. 19. If now we take a square array of elements such as that in Art. 12, we can form a system of n alternate numbers of the n'^ order by taking the elements of each row to form the coefficients of the units in the numbers. Let P be the product of all these numbers, so that P = (a„e^ + a,,e, + . . . + a,^e„) (a.^e^ + a,/, + . . . + a,„e„) . . . 14 THEORY OF DETERMINANTS. [CSAP. I. On multiplying out the factors on the right, If e^, e^ ... e„ were ordinary scalars the product e/, ... e, would be formed by taking n numbers from e^, e^...e,^, and any number might be repeated 1, 2...n times; but since e/^...e, = if any two units are alike, it follows that ^J, g' ... s is to be a permutation of 1, 2...W. It follows at once from the law of multiplication (equation 1, Art. 16) that e^e, ...e.= (-l)''e.e, ...e„, where v is the number of inversions in the series e^e^... e,. Thus P = e^e^ ... e„2 (- 1)" a.^a,, ... a„„ but the term under the summation sign is a term of the deter- minant of the system of elements, with its proper sign. Thus -P= I «« I ^1^2 ••• e„ = I a* I • Hence the determinant of a system of n' elements is expressed as a product of n alternate numbers linear in these elements. From this it immediately follows that if all the elements of a row are multiplied by the same number the determinant is multiplied by that number, and if all the elements of a row vanish the deter- minant vanishes. In future we shall write for a determinant of the n^ whichever of the forms order n^,. S + a,^a^. {Aj = a,-ie^ + aj^e^ + . . . + a,„e„) is most convenient. The letters i, h, j taking all the values 1, 2 ...n. 20. If the determinant is so constituted that the different factors of which it is composed do not contain all the units, its evaluation is frequently readily effected. For example, the determinant hi> 0, hl> «22. Ku a,„ «», 19—21.] EXAMPLES. 15 in whicli all the elements above the leading diagonal vanish reduces to the product a^^a^^ ••• <^m- For it is equal to the product of the alternate numbers «81«1 + <^A + «33«3 ««1«1 + «n A + ttna^s + ■ • ■ + an«e»- Since the first number contains e^ , and e^ only, all terms in the product of the remaining factors which contain e^ disappear when multiplied by this factor, so that as far as we are concerned we may suppose a^^, a^^... a„^ to vanish. The second number reduces to 022^2, and the product of the first two to a^^e^a^^e^. We may shew in like manner that a^^', a^^... may vanish, and so on. Finally the product reduces to By an interchange of rows and columns it follows that the determinant for which all the elements below the leading diagonal vanish also reduces to its leading term. 21. As another example let us consider the determinant 0, cos (a^ + a^, cos {a^ + a^ cos (a^ + ttj), 0, cos (cSj + Kg) cos (^3 4- aj, cos (ffg + ctg), D = of order n : the element in the i*" row andy"' column is cos (a^ + a,) unless i =j, when it vanishes. Substitute for the cosines their exponential values and write 6'^^'^ = a. Then D is the product of such factors as 16 THEORY OF DETERMINANTS. [CHAP. I. where E = a^e^ + a^e^ + ...+a„e„, F=-'+^ + ... +^. Thus if a,E+~=A„ a, we see that (- 2)" D = 11 (2 cos 2a, . e, - A^. Now observe that since the quantities A, depend only on the two alternate numbers E and F, the product of more than two of them must vanish. Hence expanding (-2)"D=2''cos2a,cos2a,...cos2a -2''cos2a,...cos2a S y^'"/" ' " ' " 2cos2a„ + 2"cos2a, ...cos2a S ^^ ' ' '/'-^ ^"-^ ^'' " 4 cos 2a^_j cos 2a^ Now e/, . . . e„., ^„ = e, ... e„_. ('«„-£:+ -) \ 71'' = ^2 + «„' = 2 cos 2 a„. ee A A =e e (^^-^\ i;i' Thus . (-y^ . =i-.-s-^^^^-'----^ or cos 2aj cos 2(^2 ... cos 2a„ cos 2aj cos 2aj, ' COS 2a, ... cos 2a„ ^ cos 2d, cos 2a, '* where («', k) are all duads derived from 1, 2 ... n. CHAPTER II. GENERAL PKOPERTIES OF DETERMINANTS. 1. If two columns or rows of a determinant be interchanged the resulting determinant is equal in value to the original, but of opposite sign. Let D = U{aae^ + ... + a^e, + ... +a,,e, + ... +a,„0> then, if D' is the determinant got by interchanging the j'" and k*^ columns, I)' = U (a,,e^ + ... + a^ej+... + %e^-¥... +a,„ej ; but since in addition we follow the ordinary commutative law, D' is got from D by interchanging Bj and e^ in the product on the right. This leaves the scalar factor unaltered but changes the sign of the product of the units, thus Interchanging two rows of a determinant, say the _/" and A;"", is the same as interchanging the two factors A^ and A,^ on the right: this is equivalent to an odd number of inversions, and hence by the rule of multiplication changes the sign of the product. 2. If two rows or columns of a determinant be identical the determinant vanishes. For by the interchange of the two columns in question the determinant changes sign, but both columns being alike the determinant remains the same, thus D = -D or Z»=0. 3. If each element of the ■P' row consist of the sum of two or more numbers the determinant splits up into the sum of two or s. D. 2 18 THEORY OF DETERMINANTS. [CHAP. II. more determinants having for elements of the i'" row the separate terms of the elements of the t"" row of the given determinant. For if I)=UA„ and A, = {a,^ + b,^) e, + (a,, + b,.) e, + . . . + (a,„ + bj e„ = (a.^e, + . . . + aj„e„) + {b,^e^ + ... + 6,„e„) since A,...A,...A^ = A^...{A\ + B^...A^ = A,...A\...A^ + A,...B,...A^, we have D = D^+ D^, where D^ and D^ are determinants having for elements of the i"" row in the A"" place a^ and b^. respectively. Repeated applications of this reasoning shew that if the elements of the i"" row consist each of the sum oi p elements, then the original determinant can be resolved into the sum of p deter- minants having for their i* rows the terms of the elements of the i*" row of the given determinant. The same theorem would apply if the elements of a column consisted of the sum of elements. In fact whenever a theorem applies to rows it applies equally to columns, as these can be inter- changed (i. 15). In future, when a theorem is stated with regard either to rows or columns, it is to be understood as applying also to the other. 4. The value of a determinant is not altered if we add to the elements of any row the corresponding elements of another row, each multiplied by the same constant factor. For if we add to the elements of the i"" row those of the fc'" row, each multiplied by p, the resulting determinant is A^...{A,+pA,)...A,...A„=A^...A,...A,...A,^+pA,...A^...A,...A, =A^...A,...A,...A^, the latter product vanishing, since it contains two identical factors. For brevity the operation of adding corresponding elements of two rows is usually spoken of as adding the rows. 3—5.] GENERAL PROPERTIES OF DETERMINANTS. 19 5. The theorem of the last article is of great importance in the reduction of determinants. The following are examples of its application: (i) If corresponding elements of two rows of a determinant have a constant ratio the determinant vanishes. For we have only to multiply the elements of one row by a proper factor and sub- tract them from the elements of the other when all the ele- ments in that row will vanish, and consequently the determinant vanishes. Of a similar nature are the two following theorems, whose proof presents no difficulty: (ii) If the ratio of the differences of corresponding elements in the p* and (f" rows to the difference of corresponding ele- ments in the r'^ and s'" rows be constant, then the determinant vanishes. (iii) If from the corresponding elements of i + 1 rows we form the i* differences and from the corresponding elements of m + 1 rows the m'" differences (the second set of rows being at least partially different from the first set); then, if the ratio of corresponding differences is constant, the determinant vanishes. (iv) Let D = M„ V^ ... t^ M„, V„ ... i Subtract each row from the one which follows it, beginning with the last but one. Then, if Am, = Mi+, - M,, we have I>= m,, v^ ••■ <, Amj, Av^ ... A*, Am,, Av^ ... At^ Au„_„Av„_,...At„_, Repeat the same operation, stopping short at the second row. 2—2 20 Then, if THEOEY OF DETERMINANTS. [chap. II. A\=Am,^,-Awj, B = u„ »i ... t. Attj, A«j ..At, AX, A\ .. A\ Proceed in this way, leaving out a row each time, and we see that D= Wj, v^ ... %> «23-"*2n -Wn> «n2. ««3---»H. — ^, Wa «!. ai2- ••«ln "2' «22- ••«2» W„. an2- ••«„» 22 THEORY OF DETERMINANTS. While any of the others, such as [chap. II. C&U^.-i ^1' vanishes, because the elements of the first column are proportional to those of the i* column. Thus «,|a«l= Ml, «.2---«„ And in general x. is obtained by substituting in the determinant |a «i2 ••• <^l» '^il' "^22 ••• ^2» «»1. ««••■«« we select any p rows, and then from the new array which these form select p columns, these when written in the form of a deter- minant constitute a minor of the given system. Such a minor is said to be of the p"" order. Since we can select p rows from n in n{n-l) ... (n-p + l) _^ 1.2...P "*"' ways, and p columns from n columns in a like number of ways, it follows that the given system of order n has (n^y minors of order p. 2. If out of the n—p rows which remain after the above p have been selected we take those n—p columns whose column suf3fixes are different from those selected in the minor of order p, we have another determinant of order n —p said to be comple- mentary to that of order p. 1 — 4.] ON THE MINORS AND EXPANSION OF A DETERMINANT. 25 For example, in the determinant a.j,, a.,2, fljj, aj4, a^^ %i' ^22' ^2s' ■■■ %S '^l.K ^62' «,1. a,, ^^21. »22 and «33. «34. "35 «43. ^44. C^45 a'53. a-M. «55 are complementary minors. 3. If p = 1, i.e. if we take a single element, the complemen- tary minor is a determinant of order n—1, which is called the complement of the element. This complement is obtained from the original determinant by omitting the row and column in which the selected element stands. For example, the complement of the element «„., which we denote by A^,., is «i*- ... a,. a.. ^i. Ct,-4.n ■ • • 0/iA.-\ 1.-1 » ^.-o-i ^x^ ■ • • ^^-j -^i+U ■ • "i+l/t-D "'i+lS+1 • ■• a«; This is sometimes spoken of as a first minor of the given determinant. In like manner the determinant formed by omit- ting p rows and p columns would he called a y minor; it is to be observed that a p*^ minor is a determinant of order n — 'p. 4. We may extend the iheaning of complementary minors as follows : From the array in Art. 1 select p rows and p columns, then from those that remain q rows and q columns, from those that remain r rows and r columns, and so on. With the elements in these selected rows and columns form determinants ; these will form a complementary system of minors if p + q + r + ...=11. The number of ways in whieh we can form such a system is \p\ q\ r\ ...\ ' 26 THEORY OF DETERMINANTS. [chap. III. It is of course permissible that one or more of the numbers p, q, T ... should be unity; the corresponding minor is then a "single element. For the determinants «24. %. «34. O^ss 1 a,, form such a complementary system, and there are 3600 such systems. 5. We have hitherto only considered the product of a set of alternate numbers equal in number to the number of units. Let us now consider the product («n«i + aii!«2 + • • • + o^A) • • • (ami^i + a^A + . . . + a„„ej ; this is equal to where p, q ... r consist of all combinations m at a time from 1,2 ...11, repetitions being allowed. First, if m > M, we must have repetitions in every term of the sum, and hence (i. 16, Equation 2) the whole vanishes. If m = n, we have the case of l. 19, and the sum is the deter- minant I a^, I • But if m '^m+21) ... a,„ "-.m^ e.fi. ■(2), where u, v . . .w is & combination of n — m numbers selected from 1, 2...W. Now multiply the equation (2) by the equation (1) and we obtain |aJ=:S{(-l)-' ttip. «W •••«!, a„,. -.. a„. 1; where from the nature of the alternate numbers e it follows that the two determinant factors under the summation sign are complementary minors, and v is the number of inversions in e,e, ... e,e„e. ... e„ or in _p, g- ... r, m, « ... w. 28 THEORY OF DETEEMINANTS. [chap. III. This theorem, usually called Laplace's theorem, gives the expansion of a determinant in the form of a sum of products of complementary minors. It is assumed in the above that the complementary minors are formed from the first m and last n—m rows. Since by a suitable change of the order of the rows and sign of the determinant any m rows can be brought into the first m places, this is no real restriction. 6. For examples we have = (12) (34) + (23) (14) + (31) (24) + (34) (12) + (14) (23) + (24) (31), «1. «2. «3. «4 K K K K c., Cj, C3, C4 d.> d^, d^, d. where for brevity (12) (34) = In like manner K> h ds.d. &c. <^l, «2. «3' "'i' <^6 K, K K K h "v C2 «?i> d^ ^1, ej where = (123)(45)+(142)(35)+(134)(25)+(243)(15) + (125) (34) + (315) (24) + (235) (14) + (145) (23) + (425) (13) + (345) (12), (123) (45) = «1. a^, a. d^, d. K K K ^4' e. c,, c»> C3 &c. 7. If when the determinant is divided into two sets of m and n — m rows there are n — m columns of zeros in the set of m rows, the determinant reduces to the product of the minor of the remaining m columns and its complementary minor. 5-8.] ON THE MINORS AND EXPANSION OF A DETEEMINANT. 29 Tliis is clear, for with the exception of this single minor of order m all the others vanish because they contain at least one column of zero elements. If the set of m. rows contains more than n — m columns of zeros the determinant vanishes. Thus, for example : a„ (x„ 0, K K, 0, ^K c„ C3, c, d„ d„ d^, rf. 16,, b^l d^, d^ while a., a„ 0, 0, = 0. \, K, 0, 0, Cj, c„ 0, 0, d„ d^, d^, d^, d^ ^l> ^2' ^3' ^i> ^5 8. In Art. 5 we resolved a determinant into the sum of products of pairs of complementary minors. We can however resolve it into a sum of products of as many complementary minors as we please. For we can divide up the n factors whose product is \a^\ as follows : Take the first u, the second v ..., the last w. The product of the first u factors would be of the form or ^IP. «!»• ■■«ir «21.' a^. ..a^ Ou,. a^- ••a„. SD„e,e, . p, q ... r being u numbers taken from 1, 2 ... n without repetition and -D„ a minor of order u from the fitrst u rows. In like manner the product of the next v factors would be D heing- a minor of order v chosen from the v rows. 30 THEORY OF DETERMINANTS. [CHAP. III. Lastly, the product of the w factors would be with a similar meaning for the quantities involved. Now form the product of all the factors, taking care to keep them in their proper order, and where Z)„, D^, ... D^ form a system of complementary minors of the determinant |a,.j.|. The sign of the term is determined from the number of inver- sions in p, q...r, f, g ...h, r,s...t. 9. If in Art. 5 we restrict the first product to the single factor a,A+a;A+"-+ai„e„ (1), the second product becomes A,,E, + A^,^...+A,„E„ (2), where J.^ is the complement of a^j (Art. 3) and Ej = e^e^...ej_,ej^,...e^. For we get a term of the product by leaving out each unit such as Bj in turn, i.e. by forming a determinant with the remain- ing n — 1 columns ; and since we previously omitted the {^ row of the given determinant, this determinant is A^. Now multiply the n — 1 factors which form (2) by the remain- ing factor (1); we obtain (- in««l = a^A^-aJ.^+ ... + {-iy-'a^A^+... For e^j^fij.e,... e^_^e^^^ . . . e„ = (-irx- «„=(-!/-'. gj^j = if y is not equal to h. The factor (— 1)'"' on the left is accounted for in the same way. Thus |«,,l = 2(-ira,^,. 8-10.] ON THE MINORS AND EXPANSION OF A DETERMINANT. 31 For example, c^i, a^, «3. a. = <'i a„ a,, a, -^2 «i. «8. «4 K K ^"3. K K K K K, ^, J. ''l' ^2' C3> c. <, d^, d^ d.> ds, d^ ^l> <^2' d. d. + ^3 K' K K d^, d^, d^ -C4 K h 10. In the final equation of Art. 9 A^ is got from |a,.j| by erasing the i"" row and /" column and writing the remainder as a determinant. It is however more symmetrical, and sometimes convenient, to give to J.^, a different form obtained by a series of cyclical permutations of rows and columns. In Afj remove the first row by a series of interchanges to the last place, then move what is now the first row to the last place, and so on, until we arrive at what was the (i — 1)'" row, which we remove to the last place. This introduces (i — l)(7i — 2) changes of sign. Now remove the first column to the last \ lace, and so on, ^' — 1 times, necessitating (j' — 1) (n — 2) changes of sign. In all we have introduced {% - 1) (n - 2) + ( j -\){n- 2), or {i +j) n changes of sign (an even number of changes being neglected). So that, if the new determinant is called J.'y, we have and where iaj=2(-ir""^%^;, A'.= ^tij+i> ^nJ+S- •a»n. «»1 ^,.^-1 •^y+n ^U+2 ^1"> '^ii ^-1 '^i-lJ+1 • '^i-lJ+1 • • • ''^l-ln ^i-n ■ ■ ■ '^i-lJ-1 32 THEORY OF DETERMINANTS. For example, [chap. Jtii. a,, a,, % = K c.> C3 + &. "a. Ci + &a Ci, "a K K K «.. ^a 0^3> «i «i. a. c,. c„ C3 In future we shall always write and suppose that A^j has its proper sign. 11. We may arrange the complements of the elements of a determinant in another square array, and then the two arrays Ai ^1,. , (1), \ (2), «n--- ....a. a ...a are said to be reciprocal. If now a sum be formed by multiplying each element of a row of (1) by the corresponding element of a row of (2), and adding these products together, the sum is equal to the original deter- minant or zero, according as the two rows have the same suffix or not. Namely, a,^A^, + a^A^ + . . . + a,„^,„ = | a,,| or 0, according as i is or is not equal to j. For if { is equal to j the sum on the left is the expansion of the determinant according to the elements of the *"" row, but if i is not equal to j the sum on the left is what the expansion of the determinant would be, if its z'"" and j"" rows were identical, but if the elements of two rows are identical the determinant vanishes. In like manner, if we multiply the elements of a column of (1) by the corresponding elements of a column of (2), we get and this sum is equal to \a^j.\ or 0, according as i is or is not equal toj. 12. If all the elements of a row vanish the determinant vanishes, as we see at once by expanding the determinant accord- ing to the elements of that row. If all but one vanish the 10 — 13.] ON THE MINORS AND EXPANSION OF A DETERMINANT. 33 determinant reduces to the product of that element and its com- plement; viz. if all the elements of the i"' row vanish except a^, then the determinant reduces to a.^Ai^. Thus for example, «.l. «12- •C^m = ^n 0, «22- •a^n 0, ^a,- •«3» 0, a„2- ••««n 0, 0, a,3...a3„ 0, 0, a„„ = a.. 0, a33...a3, 0- a„ 13. The theorem of the preceding article is of use in evaluat- ing a determinant by reducing it to one of lower order. If the determinant is not of the required form to begin with, it can sometimes be reduced to it. We may exemplify this by finding the value of the determinant D.— 0, a, a...a b, 0, a... a b, b, 0...a h, b, 6...0 (r), the sufiSxes denoting the order of the determinant. The elements of the leading diagonal are zero, those to the right of it all equal to a, and those to the left all equal to b. s. D. 3 34. THEORY OF DETERMINANTS. [chap. III. If we subtract each row from the one which follows it, begin- ning with the last but one, D. 0, a, a, a. b, -a, 0, 0. 0, h, -a, 0. 0, 0, 0, 0.. (r). The first column contains only one element, hence A=-* a, a, a, a... b, -a, 0, 0... 0, b, -a, 0... 0, 0, b, -a... (r-l). Regard the elements in the first row as a + 0, + a, + a... then (ii. 3) we can resolve the determinant into the sum of two. -b D^^-b a, 0, 0, 0... b, -a, 0, 0... 0, b, -a, 0... 0, 0, h, — a... 0, a, a, a... b, -a, 0, 0... 0, b, -a, 0... (r-l) (r-l). In the first of these two determinants all the elements above the leading diagonal vanish, hence its value is (— 1)'"^ a'"'. The second determinant is of the same form as that to which we first reduced Z>„ hence D^ = -bD^_,+ b{-ay-\ This is an equation of differences with constant coefficients for Z*,, its solution is 13 — 15.] ON THE MINORS AND EXPANSION OF A DETERMINANT. 35 14. In Art. 11 we saw how under cei-tain circumstances the order of a determinant might be reduced. Conversely we are enabled to increase the order of a determinant without altering its value, namely, by bordering it with a new row and column in one of which all the elements vanish except that common to the other. Thus aj= 1, 0, 0, ... X, a^^, a,„ a^3 ... = (- 1)" 0, 0, ... 0, 1 *21' '^22' ^2! y where the quantities x, y ... are any whatever. By adding on to these a new row and column we can raise the order of the deter- minant to w -f 2 and so on. 15. In the determinant D=\a^\, if we suppose only the element a^^ to vary, since on expanding according to the elements of the i"" row the only variable term on the right is the product a^, A,,^, we see at once that dD da. — -^if If among the elements of A^ only a„ is variable, we see that dA„ d'D da„ da„da^ ' Thus -^2 • ■ • P f so that B.! ... a^ The first two suffixes tell us the row and column in which the element stands, the third the determinant to which it belongs. The original determinant is denoted by i)'"'. The index in brackets tells us the order of the determinant. — _ \ 17 — 20.] ON THE MINORS AND EXPANSION OF A DETERMINANT. 39 19. We shall find it necessary to employ the term comple- mentary minors in the following sense. From the elements of i)j'"', form a minor i)^'") of order a by selecting a rows and columns. Then in Z)^'"' select y8 rows and columns, whose suffixes are different from those selected to form D^^'^\ these form a determi- nant D^^\ and so on until we take ir rows and columns from D^"\ to form a determinant D^W none of which have the same suffix as any of the preceding. Then if a + y8 + 7+...+7r = n (1), shall be called a series of complementary minors. Any one or more of the numbers a, /S ... tt can be unity or zero. 20. We shall now prove that ' where the meanings of the summation signs will be explained presently. For we have Z)'"' = n (a,,e, + a,,e, + . . . + a,„0, and if m,, = a^^e^ + a,,/, + . . . + «,„/„, i)<'" = nK + z*,, + . ..+«.,) (2), the product containing n factors. We shall obtain a term of the product on the right if we take a factors such as w^, /3 factors such as Ui^...ir factors such as u^^, provided the equation (1) is satisfied. But from the definition of a determinant this product of factors is equal to a determinant of order n the first a of whose rows come from D/"', the next yQ from D^'"' ... the last tt from D^'"'. Expand this determinant in the sum of products of complemen- tary minors of order a, /3 . . . tt selecting the rows of the minors from the first a, the next /3 . . . the last tt, its value is then (Art. 8) with the notation of Art. 19, and the summation sign means that we are to take all the possible complementary minors. 40 THEORY OF DETERMINANTS. [chap. III. This is only a single term in the expansion of the product (2), the whole product is obtained by summing this for all values of a, /3 . . . TT which satisfy the equation (1). Thus jD'"' = ,SSX»i(")-D,(^' ...-D^W (3). 21. The number of terms in the sum S is a\^\ vr Let us compare the expansion (3) with the expansion of the multinomial (i), + i), + ...+!)/. The general term is CD^-^Df...D/ (4), where a, /3 ... tt satisfy (1) and a\/d\...'!J-l' Comparing (.3) and (4) we see that in expanding the determi- nant we replace C by %, and a, /8 . . . tt are no longer exponents, but merely indicate the order of the determinant. Hence we may write symbolically for the expansion of our determinant where in every term of the multinomial expansion we replace the coefficient by a summation sign, the number of terms in the sum being given by the multinomial coefficient and the exponents a, /S ... TT now indicating the orders of the complementary minors. Thus finally we have the symbolical equation 22. Let us make use of this theorem to expand the deter- minant D = au+^i. «S2+^.. a,. . «a3 + «8--- according to products of the quantities z^, «^ ... z„. 20 — 23.] ON THE MINORS AND EXPANSION OF A DETERMINANT. 41 Here we must write J) w = a a„ -D '"' = z., 0. 0. ^,. . . 0, 0.. • ^« Then by the above theorem = D w + tB^^-^'D^"-^ + SD,'"-'='I>/' + ...+D^. Now clearly all minors of DJ"' vanish except those whose leading diagonal is part of the leading diagonal of I)^"\ Thus D. (!)_ ;, D/' = 0A-A"" = ^.^.-^«- The corresponding minors Z)/" ", J)/" ^' . . . are got by erasing in i)j'"' the i"" row and column, the i* and A;"" rows and columns, &c. Thus D = i),'"' + Ss.D^'"-'' + 2s,s,Z>/"-'' + . . . + z^z^ . . . a„. Or if we simply denote D^'"' by Dj, ■da,, If 0, =2;„... = s„ we get ' da,M^^ i) = Dj + zS T— i + » S , .7. ■ + . . . + 2 . c?a« da,,da^^ 23. Any determinant can be written in the form D = + a„, a,, a,, , + a,, . + cf„. We may now apply the theorem of Art. 22 by supposing A = and 0, a,, ... a„ a,i, ... ra,, 42 THEOEY OF DETERMINANTS. [chap. III. Then The general term being '■ da^^da^^ Where Z)/"^' is the minor obtained from JD^ by suppressing the t"", ft'" ... r'" rows and columns, m in number. It is clear that D^'" is zero, for a term of D cannot contain n — l terms from the leading diagonal only, if it does it miist contain n. Ex. If 0, cf,, «„., = (12), &c. we have = <^u"'As"'M + aii«22 (34) + «n<^33 (24) + a,^a^ (23) + a,A3 (14) + a,A4 (13) + a A, (12) + a„ (234) + a,, (134) + ^33 (124) + a,, (123) + (12.34). As another example we may find the value of the determinant D = Cj, a, a, a ... a b, c^, a, a ... a h, h, c^, a ... a h, h, h, h ... c„ The general term in the expansion of this determinant is 2cA ... c^d;—', when G., c^ ... Cr are any m elements of the leading diagonal. But by Art. 13 J\ in-m) _ /_ JXn-OT-l '^" I n-m-1 _ ln-m-l\ Whence if f{x) = (c^ — x) {c^ — x) ... (c„ — oc), it is clear that j_ a/(6)-6/(ffl) ^ a — b 23 — 25.] ON THE MINORS AND EXPANSION OP A DETERMINANT. 43 If we write down the similar determinant of order n + 1, for which c„+i = 0, after dividing both sides by ab, we get c„ a a, a. 1, J (a) -fib) a—h If we suppose a = h, we get on evaluating the vanishing fraction in this latter determinant a determinant expression for f\a). 24. We have seen how to expand a determinant according to the elements of a row or column. It is frequently useful to be able to expand a determinant according to the elements of a row and column. This is effected by means of the following theorem due to Cauchy, I a* J = ar»^r« - 2a,A-B«, which expands a determinant according to the products of ele- ments standing in the r* row and s* column. A^^ is the complement of a^^ and B^^ is the complement of a^ in A^^, and is therefore a second minor of the original deter- minant. For every term which does not contain a^^ must contain some other element from the r* row and some other element from the 5* column, and hence contains such a product as a^^fli,, where i and k are different from r and s respectively. The aggregate of all terms which multiply a,^ is A^^ ; now a^aj, differs from a^a^ ^7 the interchange of the suffixes k and s, thus the aggregate of terms which multiplies a^^ai, differs in sign only from that which multi- plies a^^a^, that is to say, differs in sign only from the coefficient of a,,, in A^^. Hence — i^j^^ is the coefficient in question. useful for expanding a determinant For example by this theorem 25. This theorem is which has been bordered. B = a„, a„ = K I «« ■2&,,&,.,^,„ where A,^ is the complement of a^^ in | a„ 44 THEORY OF DETERMINANTS. [chap. III. By the selection of a suitable bordering we are often able to evaluate a determinant by means of this theorem. For example, let B = a?,, a.^, a^ ■■■ a, Kj, x^, ttg ... a, «!. a^, <^z ••• «■« ttj, ttj, dg ... «„ all the elements in the i^ column being a^ except that in the i" row which is x^. Then by Art. 14 i) = 1, 0, 0, 1, x^, a^, a, 1, cij, «,, a^ 1, a„ a„ x^ Multiply the first column by a^, and subtract it from the i'" column ; do this for each column, the value of the determinant is unaltered, and 2) = 1, - «!, - a^, - (Xg, . 1, x^-a^, 0, 0, . 1, 0, x^-a^, 0, 1, 0, 0, x^-a^,. Here the bordered determinant is x^ — a^, 0, 0, «„-«„, 0, 0, for which all first minors vanish except those of diagonal elements. Hence, in the theorem of this article, we must suppose i = k; if f'M-i it follows that D=f+taJ'(x;), a theorem due to Sardi. CHAPTER IV. ON the' multiplication of determinants. 1. If we have two arrays «ii. «i2 ••• «in K' K ■■■ K «2i> ^22 •••«., (1). ^1. K ••• Kn (2), and form a new set of elements c^ by multiplying each element in the i'" row of (1) by the corresponding element in the A;* row of (2j and adding the products, these elements form a new square array of m^ elements where c« = OiiK + a.2&*2 + • • ■ + a,„b^^. This array is said to be compounded of the arrays (1) and (2). 2. We shall now shew that the determinant | Cj^ | is equal to zero if the two arrays (1) and (2) are redundant {m>n); is equal to the product of the two determinants | a,,. |, 1 6^,. | if m = « ; and if the arrays are defective (m n the product on the right vanishes, for on multi- plying it out, in each term some one of the -B's is repeated and the product vanishes. (ii) If m = M since by I. 17 the B's follow the same law as the units e, ic,J = |aJ.nA {1 = 1,2. ..n) = \aiA-\K\- (iii) If m < n the product on the right is the sum of such terms as B^BB ... p q r when p, q, r ... are m numbers taken from 1,2 ... n (ill. 5). But K' K' K ■■ B,B,B^. ^mpt ^mgy Thus mp ) rngy -mr c«| = S «!,». «,,> «„ K' K' K ■■■ 2P' "2?' 21- •■• h h 7np > ma J mp' "mq) "mr ^mp! ^mq> ^mf where for p, q, r ... we are to write all possible m - ads from the n numbers 1, 2 ... to. 3. The second case of Art. 2 gives us the rule for multi- plying two determinants. We see also that the product of two determinants of the n'^ order is also a determinant of the m* order. Thus «ii ••• ^m ^1 •■• \n Cn ■■• Om nl ' ■ ■ nn 2—5.] ON THE MULTIPLICATION OF DETERMINANTS. 47 where the quantities c,.j are given by But since in either, or both of the determinants \a^^\, j 6„ | we may interchange rows and columns without affecting their value, we see that the product of two determinants can be obtained in the form of a determinant in four different ways, viz. the element c^^ has one of the four forms : where we multiply the elements of a row of | a,,. | by the correspond- ing elements of a row or column of 16^1; o^ the elements of a column of | a^ | by the corresponding elements of a column or row of |6;j.|. There are really only two essentially distinct cases: multiplying by rows, when we multiply corresponding elements of two rows together; and multiplying by rows and columns, where we multiply the elements of a row by the corresponding elements of a column. 4. We can only compound two arrays when they have the same number of rows and columns, but we can always form the product of two determinants, for by ill. 14 the order of one of them can be increased until it is equal to that of the other without altering the value of the determinant. So that the product of two determinants of orders n and m (w>m) is a determinant of order n. 6. Examples. Compounding the two systems we get the theorem I <^,Pi + ^ ?i + Ci^> ctj>i + Kq, + o^r^ I ttiPs + &, ?2 + c.^2. aj>^ + Kl^ + Cj'-s P.' P K K C3 a,, b„ c„ d. Pv ?1 = %, K, c„ d^ i^^. ?2 a,, K C3. <^8 ^4' ^' ^4' ^4 «!, &i, Cj, ^. 0^2'L^2' Cj, ^^2 «3. ^S. C3. ^3 ^4, &4, C„ C^. P,' ?1> 0, P.' ?2. 0, 0, 0, 1, 0, 0, 0, 1 (forming the product by rows and columns) o-iPi+KP-z' «i!2'i+^i?2' Cp d, <^,Pi+Kp,' "■^Si + K^,' c„ d^ <^zP,+KP^' «3?i + ^8?2. C3, d^ a,Pi + hP,' o^, c^ ^2. K K c. a„ c. or Again «2' ^2' c„ «3> ^3. ^3 a/ +^' +Ci', aA + 6,6, + c,c„ a.a, + 6,6, + c.Cg aA + ¥2 + CiC„ < +6/ +c/, a,a3 + 6,63 + 0,03 aA + ^^3 + c,03, a,a3 + 6,63 + 0,03, < +63' +C3' \ 7. Prof. Sylvester has shewn how, by the artifice of bordering the determinants as in III. 14, the product of two determinants of order n can be represented in n + 1 distinct forms. We shall illustrate this for the case n = 3. s. D. 4 50 THEORY OF DETERMINANTS. [CHAP. IV. The product of the two determinants is the determinant of order 3 : But if before forming their product we write the determinants in the respective forms a„ h„ Cj, - ttj. K' o^' ^3, &3, Cg, 0, 0, 0, 1 Pi> Iv 0> »-i P,> 2^. 0, r. P.' ^a. 0. ^•s Q . 0, 1, their product by rows is the determinant of order 4 : «sPi + ^s?i. ^si's + ^s?.' "si's + ^sS'a. ^3 Again writing the original determinants in the forms a„ 6j, Cj, 0, a„ \, c„ 0, ffig, 63, C3, 0, , 0, 0, 1, , , , 0, 1 Pv 0. 0, ?,, r^ P2> 0, 0, 2'^, r-, ft. 0. 0' ?3. ^ , 1, 0, , . 0, 1, , their product is now the determinant of order 5 : O^iFi' «iP2' "iPa' ^' ^1 «2Pl. «^2P2> ^2^3' ^2' ^2 C^sFi. Q^aft) «3P3. ^3. C3 '*, , '"n . '•3.0.0 .7, 8,] ON THE MULTIPLICATION OP DETERMINANTS. While writing the determinants in the forms 51 Oi, 6i, Ci, 0, 0, a^, \, Ca, 0, 0, as, \, c„ 0, 0, 0, 0, 0, 1, 0, , , , 0, 1, 0, 0, 0, 0, 0, 1 1, 0, 0, . , 0, 1, 0, . , 0, 0, 1, . , 0, 0, 0, p„ q^, n 0, 0, 0, p^, q^, n 0, 0, 0, p^, q„ rj their product is the determinant of the sixth order «!, \, q, , , a^, \, c^, , , ffls: ^3, Cj, , , , , , pi, g-i, r, , 0, 0, ^2, q^, r^ 0, 0, 0, jjg, ^■j, r^ This rule is interesting as giving us a complete scale whereby we may represent the product of two determinants of order w by a determinant of any order from n to 2n inclusive; it is also frequently useful in applications of the theory, 8. The fundamental theorem of Art. 2 regarding the deter- minant formed by compounding two arrays can be deduced as follows from Laplace's theorem, III. 5. We can write the determinant | c^j | in the form of the deter- minant of order (n + m), III. 14. c„ . • • "lOT , K- -K GmX- • ^mm. K.- ■ ^mn . • , 1 . . ... , ... 1 where c„ has the value ascribed to it in Art. 1. 4—2 52 THEORY OF DETERMINANTS. [chap, IV. Now from the i"" columii subtract the last n columns multiplied respectively by a^, a,^... then from the value of c^ it follows that = , K-h ... , i^,...K -a,. -a„ ..-a„i, 1 ... In the determinant on the right multiply the first m column^ by — 1 and then move the second m rows to the beginning, then (after m + m" changes of sign) our determinant is equal to ••• *ml ' -■- , «1» •• .. . ,b,,. .1,0 .. . .. ■ ,6„,. ..0,1 .. ■ Kn . „ ... a^ , ... , ... 1 Now expand this by Laplace's theorem according to minors of the first m columns. Let us find the complement of the minor For this purpose we move the rows of a's having the suflSxes f, g... up to the beginning; then move those columns of &'s which have the suffixes/, g... into the {m + iy\ (m + 2)'"'... places. This does not alter the value or sign of the determinant, and in every place where a 1 stood before, will now again stand 1. Hence the required complement is K> K- ..0 ..0 0...0 1 1 K' K 8. 9.] Hence ON THE MULTIPLICATION OF DETERMINANTS. 53 \ = t K> K- when /, g... is an m-ad from 1, 2...w. former result. This agrees with our 9. The value of any minor of order /* of the determinant \ca\, the product of two determinants la^. | and |5jjj|. say, c.= can be expressed as the sum of products of corresponding minors of order /* of the determinants \af^\ and |6^|. For the elements of Cy are got by compounding the two arrays '^plt "H2 • • • "pn «si> aj2...aj„ 6,1, h^...h„ And since these arrays have more columns than rows, it follows that G^ is the sum of w^ products of determinants of order /*, formed by selecting yu. columns from the two arrays. Thus c. =s «/(. «/r ■a,r K. K- ■K a»f. a^- ■a,r K. K- -K when i,j...r is any /j,- ad from 1, 2...n. One particular case of this we shall find presently of import- ance; namely, when the two systems a and b are identical, and when moreover f=p, g = q...k = s, so that the leading diagonal of 0^ consists of elements from the leading diagonal of Ic^]. Then we see that C, = t is a sum of n^ squares. 54 THEORY OF DETERMINANTS. [CHAP. IV. 10. The differential coefficients of a determinant C, elements Ca, -which is the product of two determinants A, B, elements a^^, b^, can be represented as the sum of products of differential coefficients of these determinants. We have AB = G (1), and Ca = aa \i + "^a ^m + • • • + ^^ ^*» • Differentiate (1) with regard to a^/, remembering that c„, c^.-.c^^ are functions of this, we get ^dA dG , dG, ^ ^dG, Multiply this equation by dB and add together all the equations which can be obtained from it by writing for^ the values 1, 2...W. Thus we get ^ dA dB dG ^ ^ dG^^ But by III. 11 all the sums on the right vanish except "ZB^J)^,,, which is equal to B, hence dG ^dA dB , , „ . Similarly we can prove the equations. d'G 1 dCf^dc'^, 1.2 d'G ^ d'A da.^da^^' 1.2.3 tZa, d'A dKv^^s, d'A ■mdapvdar {p. 9. = 1,2... d'B n), dc,^dcj^dc„ „ ■ d\J\Jh^ (w, V, w — 1, 2 . . . n), whence the general law is obvious. CHAPTER V. ON DETERMINANTS OP COMPOUND SYSTEMS. 1. If the elements of a determinant are not simple quantities hut themselves determinants, the determinant is called a compound determinant. Compound determinants are usually formed from the minors of one or more determinants. 2. The number of all possible minors of order m of a given determinant is {w„}* (ill. 1). We can form a square array with these minors, writing in the same row all those which proceed from the same selection of rows of the given determinant, and similarly for the columns. If ?i„= fi and we give to the combinations of rows and columns taken to form minors the suffixes 1, 2 .../*, we may denote that minor whose elements belong to the i* combination of rows and/^ combination of columns, by p^, and the whole system of minors will be [ «■ Corresponding to each element iu this array, which is a minor of the original determinant, we have a complementary minor of order n — m. We shall denote the complement of p^. by g^., then these form a new array, ffii ••■ ffi/* ) (2). 56 THEORY OF DETERMINANTS. [CHAP, V. The arrays (1) and (2) are called reciprocal arrays of the m^ order. Minors of these arrays formed from the same selection of rows and columns in each are called conjugate minors. The simplest instance of two such arrays is the original system and its system of first minors, viz. a a,. A A,„ ■ ■■ a„, ..A 3. If we multiply the elements of the i"" row of the array (1) by the corresponding elements of the A'" row of (2) the sum of the products is equal to A or zero according as i is or is not equal to k, viz. Pni^*i + «a^« + . . . + a,„A,^ = AovO according as i is or is not equal to k is considered in ill. 11. 4. Let _ ^ = \aj, B = \h^\ be two determinants each of order n for which we have formed the systems of /i^ elements discussed in Art. 2 ; the systems for the determinant J. being denoted byp^j, q^, those for the determinant We can form two new systems each of fj? elements as follows. In the determinant A replace each combination of the rows m at a time by the fixed selection of rows marked i in the determinant B, this will give us /* determinants which we shall denote by ^ji, ^,2 ... tiij,. In the determinant B replace the fixed selection of rows marked k by each combination from A in turn;, these deter- minants are called u^^, \^ ... Vk/x- We have then two new systems *ii • • • ^ift "^^n ' • " ^V t^i . . . t^i^ l^mi .2-6.] ON DETERMINANTS OF COMPOUND SYSTEMS. 57 Then by Laplace's theorem we have the two sets of equations ; p'il-4. ^'=PHiqki+Pk!am+--- ti2=P'iiqn+P'A2+--- Whence by Art. 3, tiiPn + ^aPsi + • t.T.Pi2 + faP22+ — =P'i^^- And hence fa Cpn?'., +i'i8?'« +•••) + 1>-2 {p^a'n +p,,q\t +...) + ••• or t^u^, + t^u,^ +...=■■ A (p\,q\, + p',,q\^ + ...). That is to say by compounding the i* and ^^ rows of the new arrays the sum is AB or according as i is or is not equal to k. 5. We now proceed to investigate properties of determinants of the elements of reciprocal systems, and first we shall examine the system of the first order. Let A = \aJ, I> = \AJ. Forming the product of these two, -41) = 10.,!, •where C^ = a^A^, + a^A^ +... + a,^A^^, and hence C^^ = il or according as i is or is not equal to k. Thus AD = A, 0, ... 0, A, ... 0, 0, A ... = ^"; 6. Any minor of order p in the system A^^ is equal to the complementary minor of its conjugate in A multiplied by A"'^. Let S±a^a,,...= 58 THEORY OF DETERMINANTS. [chap. V. and S±A^A^^...'be two conjugate minors in the two systems each of order p, and let S ±arJJ>^ ■••he the complement of So that t + a^a^^... A = = S±a/ja^.^^a^a„ (!)• We may write S + a„a,„ ... =co t ± a,p.^^ ... Now we may write S ± A^A^ ... as the determinant of order n, ■^/i > ^fk • • • ^/u J -^/« • ■ A A A A which consists of four parts. The first square consists of the elements of S ± A^^A^^ ... ; to the right of this is a rectangle of n—p columns and p rows containing the remaining elements of the/'", g^^ ... rows. The rectangle on the left below of ^ columns and n —p rows consists solely of zeros, and the square on the right oi n — p rows and columns contains I's in the leading diagonal and zeros elsewhere. Multiply this by the determinant A written in the form (1) above. Then (ill. 11) we have At ± A,,A, A, • ' "'ffM » gv * " 0, 0, . . •• «■«> «■«••• = A'"Z±a^a^... If we resolve the determinant on the right into products of minors of the first p and last n—p columns, .:t±A„A^,... 'A^-'cot±a^fl,,.. ■6—8.] ON DETERMINANTS OF COMPOUND SYSTEMS. 59 From this it follows that the ratio of two minors of the same order of the system A^ is the same as the ratio of the comple- mentary minors of their conjugates. s + ^,A. _ co't ± a^fC V-- 7. As examples of the theorem in Art. 6, we have A,,. ■K = A''-' 'V i,p+i ••• a P+l, n ^Pi--- -^pp «np+l •••«». y A A I = A"-'^' «,1 - «.p -^w+i ••■ -^„„ %^-a^ The relation ^it> -^i. = Aco «i*. «,, A., A. ^rt, a-r. may also be written dA dA dA dA , d^A da^ ' da^, da^ da^^ da^^da^' in particular dA dA dA dA d'A f^a»-i,n-i ■ da„„ da„^,„ ' da„„_i " da„^^_„_^da„. li A=0, we see that A,> A = 0, A., A. or A A, k h ~ 1 That is to say, if the determinant vanishes, the minors of the elements of any row are proportional to the corresponding minors of the elements of any other row. 8. As an example of the use of the method of Arts. 20 and 21 of Chap. III., let us discuss the value of the determinant P = I \a,j + ii\^ I , a,s and 6.^ being elements of two determinants of the 71* order ^'"'=|a„ 5'"' =16. 60 THEORY OF DETERMINANTS. [CHAP. V. Symbolically we can write )"• Now let ^j'"', 5j'"' be two determinants of order n, whose elements are 1 d^'"' „ 1 d5'"' ""~^'»' da., ' ^'* £'»' ■ d&.. ' then by Art. 5 » ^ W _ I ^«: I _ Jl_ sn 7? <"' = — Or, symbolically. Thus P = ^"5" (X^i + iiA^. But (\Bj + fJ'Ay is the symbolical expression for a determi- nant of order n with binomial elements of the form Hence, passing from symbolic to real expressions, we have the determinant equation : 1 ^a«+ /"&« 1 = I ««. I • 1 &a I ■ 1 ty8i*+ /*«;. I . Numerous other transformations of the determinant on the left can be effected. 9. Next let us consider reciprocal arrays of order m. (Art. 2.) Let A=|2;,, I, A'=|^, |. The product AA' is a determinant of order [i. whose general element is which is equal to -4. or according as i is or is not equal to h, (Art. 3.) Hence in the product determinant all the elements vanish except those in the principal diagonal. Thus AA' = A*". 8 — 11.] ON DETERMINANTS OF COMPOUND SYSTEMS. 61 It follows therefore that A is a divisor of Ai^. Now A is a. linear function of one of its elements, say a^^, hence A can only differ from a power of ^ by a coefficient independent of the elements of A. Among the combinations m at a time of the numbers 1, 2 ... n there are which contain 1. Hence there are \ elements of A, which contain ttjj, such for example as p^^, p^^ ... p^. Hence A=a;^'^, where x does not depend on the elements of A. To determine the value of x, let Kji = except when i = k, and let aa= 1. The same will be the case with the elements p^; .-. A = l, A = 1, and .-. x = \. Thus A = ^("-i)»-i, and A' = J.(»-i)™ for w» -(w-l)™-i= («-!)»• 10. A minor of order r of the system q^^ is equal to the com- plement of its conjugate multiplied by A^'^. For if we multiply the determinant S + g^^j'^s • ■ • by the deter- minant A in the same manner as we did in Art. 6 for systems of the first order, we get : ^S ± q^q,^ ... =A'cot ±PflP,^ ...; •■• 2 ± q,a,^ ... = A^'^-cot ±PfP,H ■■■ And in like manner 2 ± PflP,k ■■.=A'- ^^--^^r^coX ± qja,i, • • • 11. Let A,^ be a minor of A, with h rows and columns. From this let us form the determinant whose elements are all the minors of order m o{ A^. These last are minors of order m of A, and are hence elements of A. On the other hand, those among them which arise from the same rows or columns of -4, and are hence in the same row or column of A, also arise from elements belonging to the same row or column of A^, which is a minor of A ; al- together they form a minor M of A, which has h^ rows and columns. While by Art. 9 we have 62 THEORY OP DETERMINANTS. [CHAP. V. which gives a representation of the minors of A by means of powers of minors of A. 12. If in the determinant A we select a minor A^ of order h, and form all the minors of order m'vsx A {m>'h), which contain neither all the h rows nor all the A columns of J.^, we shall form a minor of A with n„ — {n— h)„_,,T0ws and columns, which is equal to (n-A-l)„_l .{(i-Dm-i-li-Wm-s} where J.„_j is the complement of ^^ in J.. Let us suppose that, as in Art. 11, we have formed the minor M in A' with (n - A)„,_j rows and columns, which is equal to and let us consider the conjugate minor a^ in A, i.e. that determi- nant whose elements are the complementary minors in A of the elements of M. From the law of formation of ilf this minor has for elements all the minors of A of order m, which have A^ as a minor. If a is the complement of a^ in A, it follows from Art. 10 that ' Substituting for M its value we have a = ./1„_4 . A The theorem is therefore proved, if we can shew that a is formed as prescribed. For this purpose we must remember that ftj has for elements all minors of A which have A,^ for one of their minors; to get a we have then to suppress among the combinations m at a time of the rows and columns of A all those which contain ■ all the rows or columns oi A^; thus a has for its elements all the minors of A with m rows and columns, such that they do not contain all the h rows or columns of A^. 13. Next let us consider the determinant of the system of elements ^^j, in Art. 4, calling this determinant T, so that Since «,, = p'^q^, +p,-,q^ + ..., 11 — 14.] ON DETERMINANTS OF COMPOUND SYSTEMS. 63 it follows that T is the product of the two determinants I Pit I and \q,,\, that is, by Art. 9, The value of the determinant of the elements u^^ is obtained by interchanging A and B, and at the same time writing n — m for m. Thus 14. The ratio of complementary minors of Tand Z7is a power of A multiplied by a power of B. For if T.= «n - t^ 4l ■•• hh ,u,.,= M„ Since T.= ... 0, ... 1 we have by the theorem of Art. 4 AB, ... 0, M^^, . UT,= "J ■• "> "M+l • .. u,^ 0, ..AB,u^^, . • u^^ 0, ■■ 0; «ft+iA+l- ■ '"s+vi 0, ... 0, M^^+1 =={AB)\U^„ which gives when we substitute for U U, iti= ^.^1.5^! where \ = (n-lX^_^-h, \={n-l)„-h. 64 THEORY OF DETERMINANTS. [chap. V. 15. If the determinants A and B of Art. 4> had not been of the same order we must have increased the order of one of them, as in III. 14, until they were both of order n. We shall make use of this to investigate some further properties of the minors of A and compound determinants formed with them. 16. li A^is a, minor of order ^.of ^, and if we border it in all possible ways with m of the remaining rows and columns of A, we get the elements of a new determinant M^ of order {n — h)^^, whose value is (»-ft-i)™ |(n-''-l)m-l For we have A = ttu • •• a^, 0, .. . »2t • •• "a, 0, 0.. . a»l • • a^, 0, .. . ttwu- •• ^h+a> 1, .. . C^W21 • •• ^h^Skl 0, 1 .. . „, ... a^, 0, ... 1 Now let us write .4^ and A for A and B in the theorem of Art. 13 and combine columns instead of rows (m is supposed less than h). Each combination m at a time of the first h columns of A will give a row of T, of which only a single element does not vanish ; the value of that element is -4^, and it will lie in the leading diagonal. The number of such rows is A„. Each combination m at a time of the columns of A taken from A. — 1 of the first h columns, the last being replaced by one of the other columns, will give a row of T, in which, besides h^ elements of order h which have no influence, there will hen — h elements of order A + 1 which will be the minor A,^, bordered with a row and column of A. The first A — 1 columns of this combination remaining fixed while the last varies among the last n — h columns of -4, we shall get n — h analogous rows in T, which will give in the diagonal of T a square of elements consisting of A,^ with the simple border. The same will be the case for each combination h — l at a time of the first h columns of A, and the determinant of elements with simple border will appear h^^^ times. Similarly we should have 15 — 18.] ON DETERMINANTS OP COMPOUND SYSTEMS. 63 the determinant of elements with a border of k rows and columns repeated A„_, times, and hence T= A,'^ . Ml^-' . M^-^ . . . Mjo (1)^ while, by Art. 13, Hence, if we admit the law, ][f _ _^ (n-'i-l)*-! ^(n-7i-l)i-2 (which is true for Jc=l, for then M„= A^). Substituting for ilfj, Jfj ...M^^, the exponent of J ^ is ^,„+ ("-^ - l)i ■ ^,„-i + {n-h-l),.h„^,+ ... + (n-h -!)„_, . h, ; if we add (n — h — 1)„ to this, by a known property of binomial coefficients it becomes [n — 1),^. Similarly the exponent of A is = ('^-i)^.-(«-A-iU- Thus from (1) and (2) ][f = jj^ (n-A-l)m ^ ()i-A-l)m-i 17. Another way of stating the theorem of Art. 16 is the following : If A^ is a minor of order h of A, and we form all the minors of A with m rows and columns which have it as a minor, we get the elements of a new determinant of order (n — h)^_,^, whose value is A {n-h-l)m-h j{n-h-l)m-h-i 18. The particular case of m= 1 is so easily stated that it is of advantage to give it here. The elements of the new determinant are of the form a„ (i, k = l, 2. ..n-h), and Id =A""'"'.^. This theorem and the theorem of Art. 16 are due to Prof. Sylvester, the proofs here given are due to M. Picquet. s. D. 5 66 THEOEY OF DETERMINANTS. [CHAP. V. 19. Another modification of the theorem of Art. 16 can be obtained as follows : Let us return to the determinants A, A' of Art. 9, and form a determinant M, with the minors of A , ' 1 71— A of order n — m; this is a minor of A' of order (w — A)„^_j. The conjugate minor in A has for elements those minors of A of order m complementary to those of M-^, and hence all those which have A^ as a minor. This is precisely the determinant of Art. 17. Whence the theorem can be stated as follows : If A^j, is a minor of A of order n — h, and if we form a deter- minant Jfj with all the minors of order n—m of - Ii) such that neither all their rows nor all their columns belong to A^_,^, which in A therefore overlap A^_j^ or belong altogether to A,„ these form a determinant N of order n^— (n— A)^„ which is equal to A {n-h-l)m-7t A[n-l)m-in-h-l)m-h h ' ' First notice that this is essentially different from the theorem of Art. 12, applied to A,^. There the determinant is formed with all the minors of the same order of A with more elements than A,,, and which do not admit all the rows and columns oi A^. Here the determinant is formed with minors of the same order of A with fewer elements than A^_^, and which do not admit all the rows and columns ^„_i. To prove the theorem it is sufficient to consider in A' the minor iV complementary to a^ in A or to 31 in A'. For W is exactly formed with regard to A^_j^ as the enunciation prescribes ; it has ^m — i'>^ — ^)m-s rows, apply to it the theorem of Art. 10, or, replacing Cj by its value, from Art. 17, iy_ A {n-h-l)m-h A{n-l)m-(.n-h-l}m-h CHAPTER VI. DETERMINANTS OF SPECIAL FOEMS. 1. When a square array is written down, it is natural to inquire what simplifications arise in the determinant of the array when special relations are supposed to exist between the elements. And looking at the figure the relations which naturally suggest themselves are those which depend on the geometrical form which the array assumes. Hence we have various forms of deter- minants obtained by supposing relationships, of equality or other- wise, to exist between elements situated symmetrically in the figure ; this shews how the notation employed has influenced the development of the theory. The moat important of these special forms are symmetrical and skew symmetrical determinants. Here the special form of geo- metrical symmetry considered is with regard to the diagonal. Elements which are situated in regard to the diagonal in the position of a point and its image with respect to a mirror coin- ciding with the diagonal, have been called conjugate: two such elements are denoted by a^ ^^^ *«■ 2. If a„= cfji, the determinant is called symmetrical. The square of any determinant is a symmetrical determinant. I'or I a,^ |'= I c„ I where c,^ = a^a^i + a^^a^^^ + ... ~ ^ki- lt follows from this that every even power of a determinant is a symmetrical determinant. .5—2 68 THEORY OF DETEEMINANTS. [chap. VI. 3. We may also suppose the determinant to be symmetrical with respect to the centre of the square formed by the elements of the determinant. Two cases arise, according as the determinant is of even or odd order. First, if the order of the determinant is 2r, we may write it in the form : i) = a^, \, Cj ... m,, w,, v^, fi^ ... 7i, /3,, a^ a,, \, c, ... m„ n^, v^, fj,^ ...7,, /3^, a. a,, K, c, . ■ m^, ^^r' K' H'r ■ • 7., ^r, a. a„ ^r, % • ■ H-r' Vr, n„ m^.. • c„ b^, a. «.. ^. %■■■ f^^ n^, m,... c,, h In this determinant add the last column to the first, the last but one to the second, the (r + 1)" to the r'^ then it becomes D = «! + «!> h+l^l ■■■ Wi + ^l. l'l> /^l ••• ^1> «, "-2 ' '*2' ''2 1 /^a ■ • "'2T^''2' "2' /*2 ■ • t~>i, -^2 «! + «!, ^ + ^1 ••• n^ + v^, w„ m^... b^, a, Now subtract the first row from the last, the second from the last but one, the r'" from the {r + iy\ then D= «2 + a2' ^2 + /?2---«2-^''2. /^2 /82 . «1 a,+a„ J, + ^, ... n, + z;„ z/, , /j,^ ... ^^ , ^^ 0, 0, 0, n^-v^, m^-n^ ... 6,-^^, a,-a, 0, n^-v^, m^-i^^ ... 6,-^,, a^-a^ 3—5.] DETERMINANTS OF SPECIAL FORMS. 69 Hence (iii. 7), Z> = dj+Oj ... W,+ Z/j a, + a, ... n^ + v^ oK-v, ... a,- a. a, — a. But if the order of the determinant is 2r + 1, it may be written in the form -D = a,, \ ... n^, M„ v^ ... ^1, a, a„ b^ ... n^, M„ I/, ... ^„ a, o„ 6, ... w„ w„ V, ... |8„ a, w,, V, ... «„ ^, r, ... z)^, V, 2„ /3, ... I/,, u.„ n, ... 6,, a, "i, /^i ... v,, i/j, «, ... \, »! By proceeding exactly as in the former case, we can shew that D = Oj + Oj ... Wj + Z/j, Mj ii-v, ... a, - a. a^ + a^ ... n^ + v^, u^ 2v^, ... 2iv, p So that when a determinant is symmetrical with respect to the centre of the square formed by its elements, it reduces to the product of two other determinants. 4. If in a determinant the conjugate elements are equal in magnitude but opposite in sign, i. e. if «« = -«!■.. the determinant is called a skew determinant. If, moreover, a„=0, the determinant is called a skew symmetrical determinant. 5. It will be useful to notice the connexion between two minors of these systems, such that the rows and columns sup- pressed to obtain the one minor correspond to the columns and rows suppressed to obtain the other. Two such minors may be denoted by P = j a^, a,, ... j , Q = \ a^„, a,, ••• 70 THEORY OF DETERMINANTS. [chap. VI. 6. If the determinant is symmetrical, i.e. if (^ik~ ^w clearly P =; Q. • A special case of this is, that in a symmetrical determinant for A^ is got by suppressing the i*^ row and A"' column, while A^ is got by suppressing the ¥^ row and i"" column, thus these determinants are of the same nature as P and Q, and are therefore equal. Thus the determinant of the reciprocal system is also symmetrical. If A is the determinant of the system dA ., ., da.. 1— = J.., + J.,.-r-^ da., '" '■' da... But = 2A.^. dA ' da.. In a symmetrical determinant A., and the like are still sym- metrical determinants. 7. If in Art. 5 we see that «« = -««. P = -a,,, -a„ ... ■{-iTQ, m being the order of the minors. Thus if m is even P=Q, but if m is odd P= — Q, 8. The calculation of skew determinants reduces to that of skew symmetrical determinants, which we shall therefore now consider. A skew symmetrical determinant of odd order vanishes, for if we multiply each row by - 1, since a.^ = - aj, this changes the rows into columns, which does not alter the value of the deter- minant. Hence, if n be its order, and hence .4 = if w is odd. ' 6 — 9.] DETERJimANTS OF SPECIAL FORMS. 71 The minor A^ differs from A^^ by the sign of every element ; hence Thus Aj^ = A.j^ if n is odd, but = — ^-1^ if n is even. Thus the reciprocal system is skew if «, is even, but symmetri- cal if n is odd. J.., is a skew sj-nuneti-ical determinant of order n — 1, and hence vanishes if n is even. We have = -A,, if n is even = if « is odd. 9. A skew symmetrical determinant of even order is a com- plete square. For if J- = 1 «* I is the determinant, since A^^ is a skew symmetrical determinant of odd order it vanishes. Hence (v. 7), if «» is the complement oia^inA^,, ■0, OTa..a^ = a^, since (x^= ol^ (Art 8). Now by (m. -i) if we expand according to products of elements in the first row and first column, since -Jj, = where i, k take the values 2, 3 ... ?i; or J = SanaaVxjQt« Thi>\^ is the square of a linear function of the elements of a row. y a„ is a determinant of order n - 2, whicli is even if n is Qhce thV^^^ ^ ^^^^ symmetrical determinant of order n will 1 72 THEORY OF DETERMIN A.NTS. [CHAP. TI. be the square of 'a rational function of its elements if one of order n — 2 is so. But when to = 2, 0, a,, Thus skew symmetrical determinants of orders 4, 6...2r are squares of rational functions of their elements. 10. Since if w = 2 the square root contains one term, when 71 = 4 the square root will contain 3, when to = 6 it will contain 5 . 3 terms, and so on. Hence a skew symmetrical determinant of even order n is the square of an aggregate of 1.3.5...ra-l terms, each consisting of the product of |to terms of A. In particular a^^ df^--- ct„_i„ is a term of ^j A, for 11. This function J A is of importance in analysis, and has been called a PfafSan by Prof. Cayley on account of the use made of it by Jacobi in his discussion of Pfaff 's problem. That value of sj A which contains aiaCtg^- ••»„_!„ as first term with positive sign will be denoted by P=[l, 2...W]. The remaining terms of P are got from the first term, by interchanging all the suffixes 2, 3 ... n in all possible ways, and giving a sign corresponding to the number of inversions. Since Ojj = — a^^ it is possible to effect the interchange in such a way that all the terms are positive. The Pfaffian changes sign on interchanging only two suffixes % and h. For if we interchange i and h in the determinant, this interchanges the i"" and F" rows as well as the i'" and &'" columns, thus the value of the determinant remains unchanged. Tf p^ is the new value of P, !^^ Hence P,= ±P. 9—12.] DETERMINANTS OF SPECIAL FORMS. 73 To determine which sign we are to take, let ns consider the aggre- gate of terms a^pg, which contain a„. Then p^, only contains terms whose suffixes are independent of i and h. The corresponding aggregate for P^ is which, in consequence of the relation 0^ = — o^i* > proves that 12. The minor a^^ is also a skew symmetrical determinant. We shall shew that 7^=(-l)'[2,...i-l,i + l,...«], or with i — 2 cyclical interchanges J'^i=\i+l,...n, 2...i-l]. it follows that the terms of the product JaL■^ Ja^ are either equal to those of a^, or equal with opposite signs. Now the product (-iy^*[2 ... i-l, i+ l...n][2...k-l,k+l ... n] and the determinant «•-'•= «22 a^.-!. a^i+i (-1)'"'. '''.-12 • • • '^f-l k-1' *i-l t+l • • ■ ''^.•+12 • • • "^.+1 /L-1' '"■i+l l+l • •• by the same number of interchanges of two suffixes, become respect- ively [k,p. q, r, s ... u, v] [p, q,r,s...v, i] and *tri)i ^t>s> ^Vr ••• ^vi And the term of the product agrees in sign with the first term of the determinant whence the theorem follows. 74 THEORY OF DETERMINANTS. [chap. vr. 13* Since we have shewn in Art. 9 that J A = a,,, ^ct^ + ct,3 Vass + • ■ • + «^i» */««»> it follows that [l,2...n] = a,,[S...n] + a,,[i...n,2] + ... + a^J2...n-l]; a relation which enables us to determine PfafEans of order n from those of order n — 2. Observe that after we have selected the suffix 1, the others are written cyclically. Hence - [l>2] = a,, [1, 2, 3, 4] = a^^a^ + a^^a^ + a^,a^^ [1, 2, 3, 4, 5, 6] = a,, [3, 4, 5, 6] + a,, [4, 5, 6, 2] + a,, [5, 6, 2, 3] + a., [6, 2, 3, 4] + a,, [2, 3, 4, 5] = '^12«34«56 + «12«35«64 + "'u'^S.'^i, + «19«««62 + «lS«46a25 + «lsa4^«69 + «14C^56'»23 + ai4«52«3S + '^14'^63««62 + ai5«62«34 + ai5»63«42 + «»15 ^=^64 <^23 + «160^230^45 + «16C^24'«53 + ^^.e^^^^'S,- = {ad+be+cf}\ In particular 0, a, —h, c - a, 0, f, e b, -f, 0, d — c, — e, —d, 14. In a skew symmetrical determinant of even order, .4« vanishes, being a skew symmetrical determinant of odd order. But (Art. 8), A. = i dA d i^ti.2....r = [l,2...n]^-[l,2...n]. Now P=[l, 2...m] = (-l)'-^[i l...«-l,i+l ...n] = (-iy-'{a,,[2...i-l,i + l...n] + ... + a^{-lf-'[l,2...i-l,t+l...k-lk + l,..n] + ...]; IS — 16.] DETEEMIKANTS OF SPECIAL FOEMS. 75 hence A,, = {- 1)'** [l,2...n] {iJc}, where [ik] is the Pfaffian got by omitting i and ^ in [1, 2 ... w]. 15. In a skew symmetrical determinant of odd order A.^ is a skew symmetrical determinant of even order, and is hence the square of a Pfaffian ; viz. A,^=[l ...i — 1, i+1 ...n]\ VA,= (-ir[l...*-l, ^+l...n] Also, since A =0, A.'=-A..A,,. Hence ^ft = [* + l •■•*»- l...i-l]lk+l...n, l...k-l}. 16. The result of bordering a skew symmetrical determinant is also of interest. The result assumes different forms accord- ing as the determinant which we border is of odd or even order. Let the original skew symmetrical determinant be ^ = I (^i. I. and let the bordered determinant be ttip, a.,,, a,2, a,3 *2)3l 1^21 > '^22' ^^23 By Cauchy's theorem (lii. 24) J) = UapA - 2a^ ai3-4,i. Now, if J. is of odd order it vanishes, and A^=[i + l...n, l...i-l][k + l...n, l...i-l]; hence, if we suppose that a^t = — (ik^> A='2.a^,a^k[i+l...n, 1 ...i-1] [k+l ..n, l...k-l] = (a„[2, 3...w] + ...)(api[2, 3...n] + ...) = [a, 1, 2...n][/3, 1, 2...W], 76 THEORY OF DETEBMINANTS. [CHAP. VI. where in the Pfaffians such expressions as dja, a^t which do not occur in the determinant are supposed to mean — Uat, — a*^ ■ But if A is of even order, D = a„p [1, 2 ... nf+-Za^,ap, (- 1^ W [1- 2 ... «] (Art. 14) = [l,2...n][7,^,l,2...nl 17. We have hitherto treated of skew symmetrical determi- nants : it is easy to reduce to these the calculation of skew deter- minants. Namely, by iii. 23, D' = D + la,,D, + %a,,a,,R, + . . . -h a„ a,, . . . a„„ , where D is what D' becomes when all the diagonal elements vanish. D. is what the coefficient of a„ in D' becomes when the diagonal elements vanish ; i*,,. the coefficient of a., a^,. in B' with the elements in the leading diagonal zeros, and so on. If all the elements in the leading diagonal are equal to x we can write this !)'=«'* + ^"-^ XD, + x"-' XD,+ ...+ x"-^ SX»„ + . . . Where I)^ is a minor of order m got by suppressing n — m rows and columns which meet in a diagonal element, the other diagonal elements being put zero, the summation extends to all m-ads in n. If m is odd, D^ vanishes, and if m is even it is a complete square. Thus, the elements being skew, so, a.j2, ffijg *31> ^32' ^> ^34 ttil, 0,i2, Cl^s, X = x' + x(a\, + a\,+ a'J -=x* + x' {a\, + a\, + a\^ + a\ + a\ + a" J 18. We can apply this last theorem to prove Euler's theorem concerning the product of two numbers, each of which is the sum 16—19.] DETERMINANTS OF SPECIAL FOKMS. 77 of four squares. Namely, we have a, i, c, d -b, a, -d, c - c, d, a, -b -d, - c, h, a P. q, r, s -2. p> - s, r - r, s, P' -? - s, - r, ?. P = {a'+V + G' + dy, = (/ + g^+rHsT Now multiply these two determinants by rows, then if we write A=ap + bq + cr + ds, B= — aq + bp — cs + dr, = — ar + bs+cp — dq,. D= — as — br + cq + dp, we get a skew determinant of the same form as the other two, whose value is whence {a' + b'+c'+d^)(p' + q' + r'+s') = A' + B'+C'+D\ If we were to effect the multiplication by rows and columns we should get another form of the same theorem; by permutating the rows and columns we get still further representations of the way in which the product of two numbers, each of which is the sum of four squares, can be represented as the sum of four squares. The total number of different ways is 48. The product of n numbers, each of which consists of the sum of four squares, can be repre- sented as the sum of four squares in 48""' different ways. 19. We have seen that the square of any determinant is a symmetrical determinant (Art. 2). Cayley and Brioschi have shewn independently that the square of a determinant of even order can be represented by a skew symmetrical determinant of even order. The process of the latter is as follows : We have A = m ' n2 • nn— 1 ' nn ■ a,. a,. a.22. -a2i---«2n. -«2„-i «„2. - «„1 •••«»„. - «„„-, 78 THEORY OF DETERMINANTS. [chap. VI. Multiply these two equal determinants together by tows, and we obtain : A' = 0, h,, L-h. hu ^' ^2s ••• h 23 2« hi' %2> %3--- where L = «ria.2 - ar2a.i + a,3«M " «,-4a.3 + • • • + C^rn-l^.n " <^rnC^.n-l. then ?,,= 0, L + t = 0. Thus A' is represented as a skew symmetrical determinant. It follows that A can be represented as a Pfaffian of the functions I. If K = 4, for example, ^11 • • • ^14 = ki ^34 + '^13 ^42 + hi L • The sign is determined by making the sign of a single term in the determinant and Pfaffian agree. If instead of interchanging columns, we interchanged rows, we should get another independent representation of the determinant as a Pfaffian. 20. A third class of determinants are those of the form a,, a,, a, ... a^_ a., a., a. ... a where all the elements in a line at right angles to the leading diagonal are the same. If the elements had been written with double suffixes we should have had the relation Such determinants have been called orthosymmetrical. Their most important property is that we can replace the elements by differences of a, . 19— 21.j DETERMINANTS OF SPECIAL FORMS. 79 For if we operate on tbe rows as we did in Chap. ii. 5 (iv), if ^ax = a^+,-a^, &c. Z> = Affl , Aa„, ••• a^ ... Aa. AV, ...AX A"-\,A''-\, ... A''-'a„ ]Mow repeat the same series of operations on the columns, beginning at the last, then Z» = Adj, AX, AX, A'a., A'-'a, A''a, A"-^a,, A"a,, An important example of this class of determinants is that where a^ is a function of k of the m"' degree in k, whose highest term has coefficient unity, the quantities a^, a^..- form an arith- metic series of the m'" order. If m = m — 1 all the elements below the second diagonal vanish, while all those in it are equal to {n — 1) !, whence the value of the determinant is (-1) 2 {(^-1)!}". If m is less than n — 1 the determinant vanishes. 21. The determinant of order r + 1, where (m + 1)^, (m + 1)^^, (m + 2)^, (m+2)^,, , (m+1),^, ... (m + lU. (m+2U... {m + 2)^^ m(m — {m + r)p^ ... («i+r)^+,. 1) ... (m-jo+1) trip 1.2...P though not orthosymmetrical, is of a similar nature ; let us call it 80 THEORY OF DETERMINANTS. [chap. VI. Divide its first row by m, the second by m + 1, . . . its (r + 1)* by m + r. Then multiply the first column by^, the second by^ + 1, . . . the last by ^ + r. Then m(m + l) ... (m + r) "■ " ^ pip + l) ... (p+r) ^ (m-l)j,_„ (m-1), ... (m-l)p^,._, (m + r-l)^_i, (m + r-l)j,... (m + r-1)^ or, if we multiply numerator and denominator of the fraction by (r+iy., Thus we obtain a series of equations by giving to m and p different values in this F... (p + r- '-V^ ■1 \ ■ in-2, P-i lr+1 F„ _(m + r-p + lX. Now V^^^ „ is the value of the last determinant in ii. 5, when we write m—p for m and 1 for d. Hence its value is unity, which gives, when we multiply the above equations together and cancel like factors, ^ (m + r), .^ ,(m + r - 1)^^^ ...{m + r-p+ 1),^, (P + rl.AP+r-lU....ir + ll^, " Another expression can be obtained for the determinant by dividing the first row by m^, the second by (m + 1)^, ... the last by (m + r)^. Then multiply the first column by p^, the second by {p + 1)^, the last by (p + r)/, the transformation gives y ^ TOp (m + 1)^ (m + 2)^ . . . (m + r\ "*■' pAp + llip + 21... [p + r)^ ■ A remarkable special case of the first form is when p = l, the value of the determinant being (m + r)^^^, i.e. .the last element in its lead- ing diagonal. 21 — 23.] DETERMINANTS OF SPECIAL FORMS. 22. If in the determinant of Art. 20 / ,7 , N (c + k + m){c + kA-m-l) ...(c + Ic+1) a^, = {c + k + m)^= i.2...m then if m = M — 1, A""'aj = 1, and we have {c + n-l)^„ {c + nX_^ ...(c + 2n-2)„_, (c + '^U, (c+ 71 + !)„_, ...{c + 2n-ll_^ 81 n{n-l) {c + 2n- 2\_„ (c + 2n - 1)„_, . . . (c + 3n- 3)„_. 23. Another class of determinants are those of the form ci^, a, ... ttj where the element in the leading diagonal is always a^, and the rest of the row is filled up with a^ ... a^ in cyclical order. The peculiar property of this determinant is that it divides by where *"' is A^a^^r A^a^^ + ...-^ A^a^. If h is equal to unity this is equal to D, by the first of equations (1), but if h is not unity it vanishes by one of the other equations. Thus D divides by Oj + a^w + •■• +«'„«" s. 82 THEORY OF DETERMINANTS. [chap. VI. Hence D = (a, + a^... + aJU{a,+ a,eo + ay + ... + a^co"''), where m is one of the roots of the equation x" —1 = 0, unity excepted. 24. Another elegant demonstration of the theorem of the preceding article is the following. If „ are the n roots of unity let P= 1, «„ <...<-' 1, »„ <...<-^ 1, «„,«/...<- Then if we write a^ + a^a> + a^ + ... + a^w""' = ^ ( there is a root — Q), this = n (A^+Ay + A,a>*+ ... + A^co^n. ■which product is equal to the second determinant. For the 2n* roots of unity being denoted by ± 1, + tOj, + m^ ... ± w„_j, the w"" roots of unity are 1, co^, a^ ... o>^_^. For example if ?i = 2 a, b, c, d = A, B d, a, b, c B, A c, d, a, b b, c, d, a where A=a^ + c^ - 2bd, B=-¥-d' -t2ac, and the value of the determinant is a*-b* + c'-d*- 2aV + 2b'd' - ia^d + Wac - 4>c'bd + M'ac. 26. If in the determinant of Art. 23 we suppose ; + ,+ "'' {r - 1)['^ {n + r -1)1^ {-Zn + r-l}^ D = e^'n (a, + a,&) + a^ + ... + a„w""') = 1. + 27. Determinants whose elements are binomial coefiScients have been discussed with great minuteness by v. Zeipel, who has given an immense number of theorems relating to -this class of determinants. One or two of these we shall now consider. 6—2 84 THEOEY OF DETERMINANTS. ' [CHAP. VI. The value of the determinant m„ n, pm^, qm^ ... tm^^ (m + ll,n+l, ip+l){'m+l\, {q+l){m + l),...{t + l)im+l)^ (m + 2l,n+2,(p+2){m+2)^, {q + 2)(m+2X...{t + 2)(m+2-)^^ (m +k)^, n + k,(p + k){m + k\, {q+ h) (m + A), . . . (< + k) {m+k)^^ is {m — n) {m—p — l){m—q— 2) ... {m—t — k + l). We must first shew that the determinant vanishes when m is equal to any one of the quantities n, p-Vl, q + 2 ...t + k-l. First let m=n, then the determinant is OTj, m, pm^, qm^ (m + l)„ m+1, (p + l){m+l\, {q + l){m+lX ... (m + ^X, m + k, {p + k){m + k)^, {q + k){m + k)^... If we subtract the second column, multiplied by p, from the third we see that the determinant is independent of p. Do this, and divide the first row by m, the second by m + 1, the third by TO + 2 ..., then multiply the first column by k, the fourth by 2, the fifth by 3 ..., then the determinant reduces to the product of m (m + 1) (m + 2) . . \.2...k . (to + k) nd the determinant (m-l),.„ 1, 0, q{m-l)^, mj_i, 1, 1, (g' + l)mj, (r+l)TOj (to -1- A; -!),_,, 1, k, {q + k){in+k-l\, (r+ k) (m + k-l\... Multiply the second column by q {m — 1)^, the third by q{m-l),+ l .TO,, and subtract their sum from the fourth column and we get the hew determinant 27.] DETERMINANTS OF SPECIAL FORMS. (™-l)*-l. 1, 0, 0, r(m-l). "^*-l. 1, 1, 0, ir + l)m^ (™ + l)*-l. 1, 2, 1, (r- + 2)(m + l). {m-\-k- 1),^,, 1, k„ \, (r + k) {m + k- 1), In this determinant multiply the second column by r [m— l)^, the third by r (m — l)j4- 1 . jWj, the fourth by r (m — 1)^ + 2. m^, and subtract the sum of their elements so multiplied from the elements of the fifth column, and proceed in a similar way with the altered determinant. Finally we reduce the determinant to the product of a finite number of factors and ("^-lu 1, 0, 0. ..0, '^^^l. 1, 1, . .0, (™+l),^t. 1, 2, 1 . ..0, (m+A;-l)^„ 1, \, \...k^, A^, In this. determinant multiply the second column by (m — l)j_,, the third by (m — l)^.^, the fourth by (m — l)t_3, &c., and subtract their sum from the elements of the first column, then each element of the first column, and consequently the determinant vanishes. Hence our determinant divides by m — n. Similarly we can shew that it divides by each of the other factors, hence it is equal to C (m — n) (m —p — 1) (m — g — 2) . . . (m—t — k + 1). To find the value of C put n=p= q= . :f = 0; then we get 2, 2(m+2\, 2(m + 2),.. 3, 3(m + 3X, 3(m + 3X.. k, k (m + k)^, k (m + ^)2 . • = Cm (m-1) ... (m -k+ 1). 86 THEORY OF DETERMINANTS.' [chap, VI. But the determinant = A; ! as we see by putting d = 1 in the last determinant of II. 5. Hence G' = l; thus the theorem is proved. 28. The determinant m„ n, pm^ ... qm^^, sm^, ... um^_^ (m + l)„ n + \, {p + V){m+l\ (m + 1) (m + 1),_, (m + /cX, n+h, {p + k){m+k)^ (m + A) (m + A;),,, {m-irr\, n + r, {p + r){m+r)^ (u + r) im + r)^_^ is equal to the product of and m,, n, {k + l){k + 2) ...r pm^, (m+l)„ n + l, {p+l){m + l\... {q-\-l){m + lX (!)• {m + k)^, n + k, {p + k)(m + k\ ... {q + k) (m -^ k\_^ That is to say, is independent of the r — k quantities s, ...u. To this determinant apply the operations of ii. 5. iv. Then in place of any element P in the j"" row we must write Then in the first column every element after the (k + 1)" vanishes, while in each of the others every element below the leading diagonal vanishes, the element in the leading diagonal of the i'" column being {i — 1). Hence if we expand the determinant by Laplace's theorem, according to minors of the first k columns it reduces to {k+\){k+2) ...r m^, n, pm^ ... qm^ m,_j, 1, |>m„+(m+l)J m^„ 0, 2 (m + 1)„ 0, 0, 0, 27—29.] DETERMINANTS OF SPECIAL FORMS. 87 which proves the theorem. For the last determinant is the result of operating, as in ii. 5.iv., on the determinant (1). The determinant (1) is known by Art. 27, and hence we know the value of the new determinant. 29. Next let us consider um^ (m + l)„ {n + l){m + l),, (p+ 1) (m+ 1)^, ... (m + r)^, {n +r){m+ r)^, {p + r){m + r)^^ . . . where k has any value from d to d + r—1 inclusive. Divide the rows by m^, (m+1)^ ... (m+r) the product on the right is the cubic determinant of the elements c„j. Thus the theorem is proved. 6 — 10.] DETERMINANTS WITH MULTIPLE SUFFIXES. 93 By multiplying A. and B. together we avoided any inversion of the ^'s and B's among themselves. If we allow for the conse- quent changes of sign we can have as many such inversions as we please, and so vary the form of the cubic determinant which represents the product. 9. The product of a cubic determinant A, whose elements are a^^, and of an ordinary determinant B, whose elements are b^, is a cubic determinant C, whose elements are c^,., where Cm = ^A^ai + bj^a,,^ + ...+bj, a,,„ . Or we treat each stratum of A as if it were an ordinary determinant to be multiplied by B, the resulting strata give G. For a = n (c,,,e,e, + c.^.e^e, + ...+ c,,^e,e„ + ... + C(nl«n«,+ c,,,6A+ • • • + Ci„„eA) = n (a,„5.e,+ a,,,5,e,+ ... a,^„B^e^ + ); where B. = b^^e^ + b^e^ + . . . + b,^e^. Since the alternate numbers B^ follow the same laws as units, this last product is a representation of the cubic determinant A by means of the units e and B. Thus G = A.e,. .e^.B,...B^ = AB. 10. It is now an obvious step to consider those functions formed of letters with more than three suffixes analogously to determinants, though when we take elements with more than three suffixes we cease to be able to picture to ourselves their arrangement topographically as we can in the case of elements with one, two or three suffixes. We can, however, conceive a set of elements with p suffixes such as ^«t...i' 71" in number, to be arranged in p sets of rectangular planes in a space of p dimensions, and forming a rectangular parallelo- 94 THEORY OF DETERMINANTS. [CHAP. VII. schemon of p dimensions. (Cf. Schlafli, Quarterly Jour. ii. p. 278.) The elements which have all suffixes the same, except i, lie in the same line, those which have all suffixes the same, with the exception of i and j, lie in the same plane, ... those which have only I in common lie in a rectangular paralleloschemon oi p — l dimensions. The product of the elements is called the leading term of the determinant of the ^"' class, which is formed by keeping the first suffixes unaltered, and writ- ing for each set of the other suffixes all possible permutations of 1, 2 ...n. To each term so obtained we give the sign corresponding to the sum of the number of inversions in the p—1 sets of variable suffixes. The whole number of terms is {n !}''"'. 11. The determinant of the p^^ class can be represented as a product of linear factors of the elements which lie in the same paralleloschemon of ^ — 1 dimensions. If e,, e, ... e. Vi, '72 ■■•'V,. he p — 1 sets of alternate units ; it is plain from reasoning similar to that in Art. 3, that the function ^ = n2a.y, ,ej6,...77, (where the sum is formed by giving to each of the suffixes j, k ...I all values from 1 to n, and then forming the product of such sums for the values 1, 2 ... w of i) is a determinant of the ^"' class and »^"' order, such as we have defined in Art. 10. 12. This definition is strictly analogous to those for deter- minants of the second and third class. A determinant of the second class is the product of linear functions of the elements of a row, one of the third class the product of n factors linear in the elements of a stratum. Here the determinant of the p"" class is the product of n factors linear in the elements of a parallelo- schemon oip — 1 dimensions. 10 — 16.] DETERMINANTS WITH MULTIPLE SUFFIXES. 95 13. It is clear that by the interchange of any two suffixes, except the first, the determinant changes sign. Also since the factors of the determinant can be written as linear expressions of each of the p — 1 sets of alternate units, it follows by the inter- change of two first suffixes the determinant undergoes p — 1 changes of sign. Thus the determinant remains unaltered or changes sign according as its class is odd or even. 14. We have kept the first suffixes in their natural order. It is however indifferent which set of suffixes is retained fixed. If the class of the determinant is odd, it is perhaps more symmetrical to keep the middle suffix unaltered ; the determinant is however not the same as before. 15. The product of a cubic determinant A, whose elements are a„-j, and of an ordinary determinant B, whose elements are b^, can be represented as a determinant of the fourth class C, whose elements c^a are given by For ^ = n (a^,,€^e^ + a,,, e^e, + ... + «,,„ e,e„ + «meA+ ). B = U {K^ri^ + b^V,+ - +KvJ- Thus clearly {In-E j,Ic,l = 1,2... n) (Inn i = l, 2 ...n) which proves the theorem. 16. The product of two cubic determinants A and B, whose elements are a^^ and 6„^., both of order n, can be represented either as a determinant of the fifth class, whose elements are or as a determinant of the fourth class, whose elements are given by ^5=n(2c^e,e,r?,) c.,, = ^^.o?> (p = l,2...n); ijkl —'^'pijpkl the order of both determinants being n. The first part of the theorem is proved as follows : (In S p,q = l,2...n; in U i=l, 2...ii.) 96 THEORY OF DETERMINANTS. [CHAP. VII. (In S r, s=l, 2.,.m; in n i=l,2...n.) Thus AB = nS a,^ &,„ e^ ej^ k^ (In 2 p,q,r,s = l,2...n; in 11 i=l, 2...W.) Which by definition proves the theorem. For the second part of the theorem we have C=n2cy,,e^6,77,. Now the sum under the product sign = 2e, {a,,B^ + a,, B, + . . . + a,„BJ ( j = 1, 2 . . . n), where ^^ = l^,^ e^ rj^ + h^,^ e^ i?, + . . . + &,,„ e^ 7?„ + ... and if we write the sum becomes B,A, + B^A,^ + ... + B,A,.. The product of this has to be taken for all values of i. It must always be taken so that in each term we have the product B^B^... B^^ ; for if two B's are repeated the term van- ishes. The value of this product is B. The remaining factors in the term are where ^, q ... r is a permutation of 1, 2 ... n. This is an ordinary determinant of class 2. Comparing this with Art. 6, we see that it is a term in the expansion of the cubic determinant ^ in a sum of determinants of class 2. All these terms occur in our product. Thus G^A.B. 17. The following theorem regarding the product of two determinants of any class can be proved by the preceding methods. 16 — 19.] DETERMINANTS WITH MULTIPLE SUFFIXES. 97 The product of two determinants of classes -p and q, whose elements are a^^ , and 6^, j. respectively, can be represented either as a determinant of class 'p-\- q—\, whose elements are or as a determinant of class p+ q—2, whose elements are Cj...im...s = Say..,, 6,„..., (i= 1, 2 ... n), all the determinants being of order n. 18. It is not difficult to see how the theorems with regard to determinants of the second class (i.e. ordinary determinants) can be extended to determinants of any other class. It is probable that determinants of higher class possess many properties peculiar to themselves, though as yet not many of these have been investi- gated. The complement of any element of a determinant is a deter- minant of the same class and next lower order. The extension of Laplace's theorem would shew how a determinant of class p and order n could be expanded in a series of products of pairs of deter- minants of class p and orders m and n — m. 19. There is no difficulty in writing down the expansions of determinants of any required class or order. The number of terms however increases very rapidly. The following are the expansions of determinants of the second order, and classes 3 and 4 respectively : 2 + (111) (222) = (111) (222)- (121) (212) + (122)(211) - (112) (221) t ± (1111) (2222) = (1111) (2222) - (1112) (2221) -t- (1212) (2121) - (1211) (2122) + (1122) (2211) - (1121) (2212) + (1221) (2112) - (1222) (2111), while for the determinant of class 3 and order 3, S ± (111) (222) (333) = (111) (222) (333) - (121) (212) (33-3) - (Ill) (232) (32.3) + (131) (212) (323) + (121) (232) (313) - (131) (222) (313) - (112) (221) (333) + (122) (211) (333) + (112) (231) (323) - (132) (211) (32.3) - (1^2) (231) (313) + (132j (221) (313) - (Ill) (223) (332) + (121) (213)(332) + (111) (233) (.322) - (131) (213) (322) - (121) (233) (312) + (131) (223) (312) + (11.3) (221) (332) - (123) (211) (332) S. D. 7 98 THEORY OF DETERMINANTS. [CHAP. VII. - (113) (231) (322) + (133) (211) (322) + (123) (231) (312) - (133) (221) (312) + (112) (223) (331) - (122) (213) (331) - (112) (233) (321) + (132) (213) (321) + (122) (233) (311) - (132) (223) (311) - (113) (222) (331) + (123) (212) (331) + (113) (232) (321) - (133) (212) (321) - (123) (232) (311) + (133) (222) (311). 20. We shall conclude this chapter with the following general theorems. A determinant of any class, all of whose elements are equal to a, except those in the leading diagonal which are equal to x, is equal to {oo+{n-V)a\{x-aY'^, n being the order of the determinant. We shall prove this for a cubic determinant, but the method is perfectly general. D = IT {ae^e^ + ae^e^ + . . . + ... +a;e,6, + ...) ^^{aEE'+ix-a^efi], where E=e^-\-e^+ ... e^, ^' = 6^+ 6, + ... +6„. Hence, since E and E' are alternate numbers, any term in which they occur more than once vanishes. Hence D =-{x- af + a{x- a)"-' I,{EE'Ue^e,} {k = l, 2 ... i-1, i+1 ... n); :. D={x-aY+ na {x - o)""' = {« + (w - 1) a] {x - a)"'' ; for Ee, ^ir-A+l •■ •■ e„ = eA •■• e,_ie,^., ... e, = {-iy-\e,... e„; ^4-l^i+l • ..e„=(-l)-V.- «»• and so j&'e^ The last theorem of iii. 25 can also be extended to determi- nants of higher class, for a cubic determinant we may state it as follows : If all the elements in the i* stratum are equal to a., with the exception of that which lies in the leading diagonal, whose value is x^, then the value of the determinant is _ /+Sa/(^J with the notation given in iii. 25. CHAPTER VIII. APPLICATIONS TO THE THEORY OF EQUATIONS AXD OF ELIMISATIOX. 1. If wo have « linear equations between n quantities .r, , .i-j . . . jr, , namely, o,., ,r^ + (7^^, .r, + . . . + a^ a-, = M, (1). the determinant J. = | a^ | is called the determinant of the system. If A does not vanish we can at once determine the variables. For if we multiply the above equations by A^.^. A,.^ ■■■A^^ respectively and add, then all the terms on the lefl vanish, with the exception of those nittltipMng .r^, which together give A i^iii. 11). Hence A.v^=u^A^. + tt^A^+ ...+ii^A^ (A-=l. -2 ...n). The expression on the right is the expansion of the determinant, obtained by writing it,, u^ ... h, for the elements of the k^ colunuL i2. It is interesting to compare with this the solution bv alternate numbers. Multiply the given system J) by e,, t\ ... c', and add; then it e, a,^+ e, 0,,+ . . . + e„ a^, = A, p, M, + <'jMj +...-T- f, M. = r. we have ^,.r, + J,.r,+ ... + J..r, = U. 100 THEORY OF DETERMINANTS. [CHAP. VIII. Multiply both sides of this equation by ^, ... A^^ A^^ ••■A„, and we get A- A-A,^, -A^A,x, = A,... A,_, A,^^ ...A, U, or A,...A„x^ = A^...A^^UA^^...A^, and writing the products of alternate numbers as determinants we get the same solution as before. 3. If in the equations (1) the quantities u on the right vanish, we have the system of n homogeneous linear equations »U*1 + «i2«2 + • • • «i»*n = «*< = (i = 1, 2 . . . w). We may regard these as equations to find — , — ... -^=i . n n n Taking any w — 1 of the equations, by Art. 1 we can determine the ratios. These values, if the equations are consistent, must satisfy the remaining equation. This condition is ^ = 0. For if we multiply the equations by A^^, A^j^ ... A^^, as before and add, we get ^'J a^ 1 = 0. If then the equations are to be satisfied by other than zero values of the variables we must have ^=0. If this be true any one of the equations is a consequence of all the rest, viz. we have u^A^, + u,A^+ ...■+ti^A„^ = 0. Where the w's now stand for the linear functions, that is to say, any one of the m's is expressible linearly in terms of the remaining ones, provided the quantities A^ do not all vanish. 4. If the condition of the preceding paragraph holds we have ^1 _ _^ _ —3s-. For if we substitute the values x^=XA^ all the equations except the k*^ are satisfied by iii. 11, and the A"" is also true since J.=0. 2—6.] ELIMINATION. 101 5. Returning again to the equations of Art. 1. Any new linear function v of the no's can be expressed linearly in terms of the u's. For if v = \x^+b^x^ + ...+b„x^, w„= a„,x, + a„x. + . . . + ax., we may regard these as n + 1 equations between the n+1 quan- tities 1, x^, x^... x^ Hence, by Art. 3, we must "have V, K, h - K u„ a,„ ffij, ... OS,, or — Av- ^n, a«, a„2 • ■ a„« 0, h, \ . • K «i, a„, a,, . • «!» = 0, w«. «»i. «„2 ••• a« 6. If we have between n variables x,^, x^... x^, the m equations «u^i + «i2^2 + - + aiA = where m is greater than n. Then if these equations are to be true for other than zero values of the variables, if we take any n of them their determinant must vanish by Art. 3. This condition is represented by a„, «X2 ••• «i, a„, a ... a = 0; which means that each of the system of m^ determinants, got by selecting any n rows of elements from the array and forming a determinant with them, is to vanish. The expression on the left is frequently called a matrix. 102 THEORY OP DETERMINANTS. [CHAP. VIII. 7. The system of linear congruences (mod.^), first considered by Gauss, has been solved as follows by Studnicka. Let A=\a,^\, and let ^^ be the greatest common measure of the numbers Then, as in Art. 1, we have for all values of k from 1 to n, A 1 - '^.^ - K^u + '^2^2. + • • • + «»^ J (mod. p). Uk Uk The advantage of the rule is that if we observe that one of the minors of a column is unity, or if two of them are prime to each other, then, for that column, g^ = 1. 8. The solution of the system in Art. 1 assumes different forms according to the nature of the coefficients a^^. If ««=-«« and a„=0, so that the determinant of the system is skew symmetrical ; first, if n is even, if we multiply the equations by \2...k-l, k+\...nl [Z...k-1, k^-l.-.n,!} ... {l...k-l, k+l.-.n-l], and add, the coefiicient of aj^ is a,,[2...7c-l, k + \ ...m] + a,,[3...i-l, k+l ...n, 1] + ... +a^^[l...k-l, k + l...n-l] = -[k,l...k-l, k + 1 ...w] = (-l)*[l, 2...W], while the coefficient of x. is -[i, l...k-l, k+l---n] = 0. Thus (-!)'«, [1,2 ...n] = u^[2...k-l, k+1 ... m] + u,[S...k-l, k + l...n, l]... + u^[l...k-l, k + l ...n-l]. But if n is odd, then A=0 (vi. 8) and x^,x^...x^in general are infinite, but bear fixed ratios to each other. If however u,A^^ + u^A^,+ ...+u^A^=0, or M, [2 ... w] + wJ3 ... n, 1] +...+i{„[l, 2 ... 7i-l] = (vi. 15), one equation of the system is superfluous, and the system of the remaining equations can be solved as above. 7—9.] ELIMINATION. 103 9. In Art. 3 we have the first example of the process of elimination; namely, we have found a condition, independent of the variables, which must hold if a certain given number of equations are to exist between these variables. When r homogeneous equa- tions hold between r variable quantities, (or what is the same thing, r non-homogeneous equations between r—\ quantities) it is always possible to establish an equation i? = between the co- efficients of these equations alone. Then B, is called the resultant or eliminant of the system of equations. When the equations are two in number the most direct process is Sylvester's dialytic method. Let the two equations be = a„ + a^x +a^x^+ .. Q = \ + \x+\a? + If we multiply the first equation by \,x, x^ .., a;""' we get n — 1 new equations, and from the second by multiplying by 1, x, x^ ... a;"*"' we get TO — 1 new equations, viz. we have now the system = a^ + ajX + a^x^ + ... = a^x + a^x^ + . . . = a.oi? + ... ...+a^x- I .(1). = ^-o + h^x + h^a? + ... = \x + l^x"" + ... = \x' + ... oim + n equations satisfied by the same values of x as the given equations (1) and linear and homogeneous in the m+n quantities Xj SCf it/ • • • y3 • Hence, by Art. 3, the determinant of the system must vanish, or R a„ = 0. 104 THEORY OF DETEEMINANTS. [chap, vin; A determinant of order m + n. Since there are « I rows of a's, and m of b's, the resultant is of order n in the coefficients of the first equation, and of order m in the coefficients of the second. 10. If the coefficients a„, a^_„ a^_, ... \, 6„.^, b^_^ ... are functions of y and z of degrees 0, 1, 2 ... , it can be proved that the resultant is of order mw in y and z. This will be the case if every term in R has the sum of the complements of the suffixes equal to mn. If we change y and z into yt and zt respectively, the value of R is now R' = ^ j.m ^ j.m—1 ^ 4.7Y> ajr, af-' bf, \r\ bf-' ... bf, bf-' ... Observe that the separate elements and therefore each term of R' is multiplied by a power of t equal to the complement of the suffix. Now, multiply the first n rows by r-\ r-' ... t, 1, and the last m by r-', r-^ ...t, 1. Then R' is multiplied by a power of t, whose exponent is m (m— 1) n (n — 1) 2 "*■ 2 ■ But now the first column of R' divides by f +"~', the second by r'"*""^, and so on. Thus .S'-^i? is equal to a power of t whose exponent is (m + n) (m +71— 1) m(m — 1) n{n — l) 2 2 2 Thus every term in R' must divide by f"', which proves the theorem. Functions, such that the sum of the suffixes, or of their complements, of the elements in each term is constant, are some- times called isobaric, and the constant sum is called the weight. 11. We may consider the question in another way. If (x) = 6„ + b^x + b^x" + ... + by = &„(*-/3,)(a;-/3J... (*-/3„) (1) = mn. 9— ll.J ELIMINATION. 105 is an equation whose roots are yS^, /S^ ... ^„, the function f{x)=u=a„+a,x+a^x^+ ... +ajc"' (2) has n values corresponding to the different values of x given by (1). These n values are the roots of an equation of the w'" degree, which we now proceed to find. Multiply the equations (1) and (2) by the same powers of a; as in Art. 9, and we have the m + n equations = a„ — M + a^x + a^ + . . . = (a„ — w) a; + a^o^ + . . . (a„ - m) a" + ... 0= 6„ + &ja; + 6,a!'+ ... = 6(,a; + h^o^ + . . . 0= Eliminating between these the quantities we get a;™^"-' ... a;, 1, a„-u, a^ . a„ — u. K, K K ... k, b, ... = 0, an equation of the n'" degree to find u, the roots of which are /(/3J,/(/8,).../(/3J. The product of the roots being equal to the constant term (- ir&:/(/3j/(/3.) -/(^J = (- ir-R. where R has the meaning in Art. 9. Thus In the same way we may shew that if a, ... a,^ are the roots of (2). 106 THEORY OF DETERMINANTS. [chap. VIII. 12. If the two functions (f> and/ of the preceding article are a function and its differential coefficient, then R is called the dis- criminant of the function, and its vanishing is the condition that the function should have equal roots. If f(x) = a^ + a^x + a^a^ + . . . + ajxf = a„{x- aj (x - a,) ... (« - aj /' («) = ttj + 2a^x + ... + na^x"'^ ii = <-y(«.)/'W-/W a^, 2a^, Sctg .. having n rows of the first, and n — loi the second kind. If we multiply the last row by n, and subtract it from the n'", this becomes ... 0, -Jia„, - (n-1) a^ ... -£i„_i, 0. Thus the determinant reduces into the product of a„ by a determinant of order 2n— 2, which we shall call A. Also / (a J = a„ (a, - aJ (a, -a,) ... {a, - a„) f (aj) = («2 - «i) «n (o'a - Os) • • • («2 - O ••• /' («.)/'(«.) • • • /'(«J = (- 1) "^ a\ ? (a„ a, . . . aJ where ? (cZj . . . a J means the product of the squares of the differ- ences of all the roots. Thus A = (-l) ^ ar?(a„ a, ...«„). 13. The artifice employed in eliminating x between two equa- tions may sometimes be employed for the case of more equations than two, as in the following examples due to Prof. Cayley. 12, 13.] ELIMINATION. 107 Let x + y + z = 0, x^—a, y" = h, z^ = c; multiply the first equation by 1, yz, zx, xy, and reduce by means of the other three, then we get x+y + 2=0 xyz + cy + hz = xyz + ex +az = xyz+hx + ay =0, whence, eliminating xyz, x, y, z, we get ., 1, 1, 1 =0. 1, ., c, b 1, c, ., a 1, b, a, . Or if we multiply the equation by x, y, z, xyz, and eliminate 1, yz, zx, xy, we get ., a, h, c =0. a, ., 1, 1 b, 1, ., 1 c, 1, 1, . Again, if we are given the equations a; + 2/ + 3 = 0, x^ = a, 2/^ = 5, s' = c, if we multiply the first equation by X, y, z, y'z\ ^:V, xY, a?ys, yhx, z'^xy, and reduce by the last three we can eliminate a^, 2/', < 2/^, z!«, «>y, ooy^z\ yzV, zxY between the resulting equations, giving = 0. 1 b a 1 1 1 Other forms of the resultant can also be obtained. 1 1 ' • 1, 1 • • 1, 1 , - • ' f 1 1 a. I, ' 1 1 . J c 1 1 108 THEORY OF DETERMINANTS. 14. The resultant of the quadric u = a^^x^^+ ...+2a^x,x^+... = 0. [chap. VIII. _^ (1), and of the n — 1 linear equations ^1 = Cna^i + c^,x^ + ...+ c,^oo„ = •(2) V.= c„-n«'i + c^iA + ■■■+ c„_i,a;„= can be readily expressed by determinants. By Euler's theorem for homogeneous functions we can write " the first equation in the form du du ^ dx^ ^ dx^ + -'.£ = 2-=o (3). Then if in equation (3) we do not consider the variables implicitly contained in the differential coefficients, (1) and (2) being n equations, between x^...x^, (3) must be identical with >-A + V2+-+ViVi = W by Art. 3. Equating coefficients in (3) and (4) we must have «ii«i + ai^a?, + . . . + a,„a;„ = \c^^ + \c^, + ...+ ^.^c^.^i a.l«l + ^n2<«^ + • • • + «»,A = \Ci„ + \,c,„ + . . . + ViC„_i„ the equations (5) together with (2) form a system of 2w— 1 equa- tions between x^, x^... x^, \,\... \_^, hence their determinant must vanish by Art. 3. Thus = 0, '^M •■• ^»K> ^m •■• ^n-m the blank space being filled with zeros. This result is due to Versluijs. a,^ and a^ mean the same thing, viz. half the coefficient of X.X.. t k 14, 15.] ELIMINATION. 109 15. If we seek to solve the system of equations x + y = a x^ + y^ = ¥, we do so by establishing the new linear equation x-y^±j2¥-a'. Following up this idea Baur has solved the non-homogeneous system of an n-ary quadric and n — 1 linear equations between the variables ; viz. let the system be + 2a^x,x^+... = u (1), + Ci„ «„ = 2/i + C2„ «„ = y. On ^'i + • •(2) + ''»-l»*n — Vn-l- Then we wish to establish a new linear equation c„.a'i+---+c„A = 2/„ (3), so that if we determine the values of a?, . . . a;„ in terms oi y^...y^ from (2) and (3), and substitute their values in (1), the result shall only contain y^ in the form y„^ We are to have then y'=y:^^Ky.y. ii^=^' 2...n-i) (4). Now if 0=\c,,\ we have Gx, = C,,y, + G^,y, + ... + G^,y^ (5). Hence, differentiating (4) partially with respect to y^, we get _ du dx^ du dx^ r '^ ^3n y''~ doc^' d^^'^ dx^' dy^ '" dx^ ' dy„ ' _-^du or, by aid of (5), if Cu- Cl2 • • Ci,. C^i. C.2 • •• %n ''n-11' ^7.-12 • • Vm «1. «2 ■ • -*«„ .(6). 110 THEORY OF DETERMINANTS. [chap. VIII. Substituting for the differential coefficients their values we determine the form of the equation (3). We have still to determine the value of y^^. To do this we introduce the n(n—V) quantities e,i, Cjj ... gjj ^21 > ^22 ■ • • ®2l sueh that and hence where Thus e^i «u + er,a2. + • • • + ^A, = c,,; ^ = I a,, I . ^11' ^12 ••• ^1, ^n-l 1 > ^m-1 2 ■ ■ • n-1 n Now from the product of (6) and (7), = Gy. (7). • ■• c. D D 7? ■)/ ■'-^n-ll' •^ti-12 ••■ ■'-'ii-ln-ll .711-1 where 5„ = c,,e,. + c^e^, + ...+ c„e,„, + ... ^' ''rK ^r2 ••• "« c.i> a„, a,2 ... a„ <^.n ' ''nl ' <^.,2 ■ • • O^ni = ^B.. ■(8), 15, 16.] ELIMINATION. Ill On the right-hand side of (8) all the quantities are known from (1) and (2). Thus Gy„ is known; substitute its value in the left of (6) and we have the required equation (3), which with the equations (2) forms a system of n linear equations sufficient to determine the quantities x^ ... x„. 16. The equation a.. = (where a^^ = a^) formed by taking X from each of the leading elements of a symmetrical determinant is of considerable im- portance in analysis. The following proof that its roots are real is due to Sylvester. If we denote the left-hand side of the equation by ^ (X) we have 4>{-x) = a,,+ \ a,, .. • a,„ «2i. a,,+ X.. ■ a.n J a„i, O^na •• • a„„+\ and hence <^(\) ^ht-V ^k*-2 ••■ «o> «*. O^m ••• + ...+af -a„ •«t4-i ••• Vi. a, ... .(4). If now x^^, a;^2 ... a!„ be roots of the equation (1) we have the n — lc identical equations = 4>u fe) + h^r" + h^:*""' +■■■ +K-A''^ From these, by aid of (3), eliminating h^ .... h^__^ we .get Or = = i^ 1 a; a;" .. fl^K Expand according to the elements of the first column and then multiply up by a^^, and we get where c^... are independent oip. This equation is satisfied by all the roots of (1), and if x^^ ... a;,, be the n — k roots of greatest absolute magnitude, when p increases indefinitely the remaining roots of (1) are by the last equation those of 4>,(x) = 0. S. D. 8 114 THEORY OF DETERMINANTS. [CHAP. VIII. Hence, if x^ is the least root of (1), a;,, x^ tlie two least roots, .a;,, x^...Xj^ the k least in absolute magnitude, then x^x^ = a, lim. {-^^^ a:^a;,...a;,= (-l/rt„lim R, 'k, p To establish this rule completely as one of practical utility it would be necessary to shew, for instance, that x^ lies between two successive convergents, obtained by taking two successive values of J), and that these convergents approached a;^, and did not recede from it. The method has been extended by Fiirstenau and Nagelsbach to the case where the roots are not all unequal, and also to the case of imaginary roots, but the discussion of these points must be omitted here. CHAPTER IX. BATIONAL FUNCTIONAL DETERMINANTS. 1. If we have a series of n quantities x, y, z ...u,t Vfe shall denote the product of all the ln{n- 1), differences obtained by- subtracting from each number all that follow it, by ^{x, y,s ... u, t). So that ^^{x,y, z ...u,t) = {cc-y).{x-z) ... (x-t) iy-z) ... (y-t) {u-t). This function f ^ {x, y, z . ... u, t) is an alternating function of all the quantities x,y,z...t; viz. on interchanging any two of these it changes its sign, but not Its absolute magnitude. It is thus of the nature of a square root, having two values equal in absolute magnitude, but opposite in sign. This is conveniently indicated by the index ^. The product of the squares of the differences will be denoted hj l^{x, y, z ... u, t), and is a symmetrical function. This notation is Sylvester's. 2. We have aj-S a;?-' ... x, 1 2/"-', y--' ...y, 1 r t 1 : ^* {x, y,z...t). For the determinant on the left vanishes if any two of the quanti- ties x, y ...t become equal, because then two rows become identical. Thus the determinant divides by the difference between each 8-2 116 THEORY OF DETERMINANTS. [chap. IX. IS as" ' ?/""* ...u.l, wticli is also a term in ^^{x ...t) with its Thus the theorem follows. pair of the letters, being a rational function. Hence it contains f^ {x, y-...t) as a. factor. But the leading term in the determinant proper sign. 3. Every alternating function oi x ...t divides by ^^ (x ... {), for on interchanging two variables the function changes sign, and hence vanishes if they become equal, thus it divides by their difference, and therefore hy ^'^ {x ... t). 4. If /.(«) be a function of the i* degree in x, the coefficient of whose highest term is unity, we have /«-.(^). fn-.{^) • •• /i W, 1 I = ?*(«, 2/ ... <). /„-i(2/)> fn-M ■■■fM' 1 f.-M /»-.(«) -/.(O. 1 For if we subtract the last column, multiplied by a proper number, from the last but one, the elements in this column become a;, y ...t. Now multiply the last two columns by the proper numbers, and subtract their sum from the last column but two, the elements of that column now become m?, if ...f. Proceed in this way and we reduce .the determinant on the right to that in Art. 2. If the coefficients of the highest powers of x were not unity, the determinant is equal io i^ {x,y ...t) multiplied by the product of the highest coefficients in the. separate functions. Forfixample, if /» = ^ l)...{x-i + l) i\ 2/n-l' lln- K^{x,y...t) {n-\)\ (n-2)! ...2!' The denominator can also be written 2"-^3"-' ... {n-2y. (n-1). 2—6.] RATIONAL FUNCTIONAL DETEBMINANTS. 117 we see by the theorem for multiplying two determinants (iv. 3) that /xk)>/..W-/„(^.) 1. c.(-yj, c,(-2//...(-2/,r l, c,(-2/,), c,(-2/J... (-y,r- = C?^(2^i. 2/2 •■• 2/J. where (7 is the product of all the binomial coefficients of order n-l. For the elements in each column of the determinant are multi- plied by that power of — 1, which is introduced by moving the column from its place in 1^ to the place it occupies. Thus If X, = 2/j this gives us ^ (0;^ . . . x„) in the form of a determinant. 6. We may also give still further determinant forms to the product ?* (a-,, x^ ... x„) ?^ (y„y.,... y„). Thus ^i(x^,x^ ...x„)^(y^,y^...y„) = ... 1 yr - 1 118 THEORY OF DETERMINANTS, where if we multiply by rows a^i2/*-l Or if we multiply by columns c,* = <~'2/r' + <''yr" +■■■+ <''y:'^- If we put x^ = 2/( and s^ = xl + xl+ ...■\- x^ we get [chap, IX. ?K, a;, ... a;J = *2n-3' ^2n-4- •• ■ ^m- ^0> *1 ••• ^n-1 an orthosymmetrical determinant. 7. A more general theorem is the following. Consider the array x^~\ x^-' m-l „ m-2 1 ajj, 1 m-l )n-2 1 where n is greater than tn. Compound it with itself, we get a determinant of the m"" order which is equal to the sum of the squares of the n^ determinants, obtained by taking any m different rows in the array. The determinant has for elements Hence, by aid of Art. 6j we get So. h- ^m-V *m • • *2«l-2 where x^,, x^ ... are any m of the n quantities x^, x,... x„. 6^9.] RATIONAL FUNCTIONAL DETERMINANTS. 8. We have clearly by Art. 2 119 ic", a;""' ... X, 1 1 ' 1 . . • fj , -•■ = r*K, a, ...«„)/(*) where / {x) ={x- a J (a; - a,) . . . (x - aj = as" - p^x"-' +^^a="-^ - . . . + (- l)"-* ^„,a;' + . . . Equate coefficients of x* on both sides and we get ^2 > "2 «! ... «! , «! ... 1 i+1 „ i-1 -1 ^,^ is the sum of the products ?i — i at a time, without repetition, of the quantities a^...(x^, 9. We may write the first equation of the preceding article in the form „ n-1 ^ n-~l „ n-1 «,«-! A «!. "2 ••• ««> ^. 1, 1 ... 1, 1, 0, ... 0, 0, 1 and similarly we have = (-ir?^(«„a, ...aj/(x), a„", 0, y" „ w-1 _, 71-1 -, n-1 /\ -,n-l fli, a^ ... a„, 0, ^ 1 ... 1, 0, 1 0, 0. 1, = (-irr^(«x-o/(y)- Form the product of these two determinants by rows, and we have = -J:(a„ a,...aj/(a;)./(y), ^n> «2.-l • ■ s„, X V-1' ^2u-2 • ■ 5«-l. a;"-' Sn, «n-l • • «o> 1 f, f-'- . 1, 120 THEORY OF DETERMINANTS. [chap. IX. from whicli by equating coefficients of tte powers of x and. 3/ we get a number of theorems, s^ is now the sum of thfc r'" powers of the roots of the equation /(a;) = 0< 10. We may extend the theorem of Art. 8 as follows: the value of the determinant x:^\ <^^ ... a;,, 1 a,'^', a."^ ... a., 1 „ n+r-l „ n-h— 2 ^ -i which is of the form of that in Art. 2, consists of three parts. First the product of all the differences of all pairs of the quantities x^ ... x^, i.e. §■ ('"i, ..• '"^t which by Art. 2 is a deter- minant. Secondly, the difference of all pairs of the quantities a^ ... a^, i.e. g! ("i .•• "J- -^^^' lastly, the product of all such quantities as /(a;.) = (00, - aj {x, -a^) ... {x, - a J = x: - p,xr + . . . -h (- vr-^'Pn-A + • ■ • Henee its value is •• 00^, 1 xr\ xr^ x„ 1 ?*(«, ...«n) /(«>,) -/K).- M.ultiply the i*" row by f{x),, and then equate coefficients of iCj" . x^. x^ ... , and we get the theorem : If -0»,», /'n-u+r-a • • • Pn-v. Pn-v+r-l* Pn-v+r-S '"Pn-v ^^(a^, ff, ... «„), 9—11.] RATIONAL FUNCTIONAL DETERMINANTS. 121 where p^ is tlie sum of tlie products ^ at a time of Kj ... a„. If k is negative or greater than n, p„=0, Pa = l' 11. Let us consider the determinant D = 1 1 1 1 1 1 x,-a^' «2-«9 <^i-a„ 1 1 1 Multiply the t"" row by /(a;J = M. = (x, - a,) {x. -a^) ... (x. - a„), we get "A ...U„D: aj, - a. The determinant on the right is an integral and alternating function both of the quantities x^ ... x„ and of a, ... a^. Hence by Art. 3 it divides by ^^{x^, x^ ...a;„)p(aj, a,... a J. Comparing the orders of the determinant and this product we see they are the same, hence the additional factor is numerical only. To determine it, put x^, x^ ... x„ equal to a^, a^ ... a„ respectively, all the elements except those in the leading diagonal vanish, and = (^ 1)"- (a,- a) ... (a._, - a,) (a,- a^J ••.(«,- «„) when x^ = a^, thus the determinant reduces to ti(n-l ) (-1) ^ ^{a„ ,-a„y And thus (-1)" d^D dxj^dx^... dx^ = B. 126 THEORY OF DETERMINANTS, We shall now sliew that [chap. IX. B r 1 1 01,- a, 1 1 D~ 1 1 «ii-«2 *2-«„ 1 1 1 «!„ - a, «;„- a, x„ - a 71 "« Where { } means that the function on the right is to be formed like a determinant, only all the signs are positive instead of alternating. Multiply the i*^ row of B hy M.^ then (u,u^...uJB-- .(1). The determinant on the right is. an integral and alternating function, both oi x,,x^...x„ and of ofj, «„ . . . a„, hence it divides by If the quotient is ^{x,,x^... x„), this is symmetrical with regard to each of the variables, and of order n — 1. Thus Now, by repeated use of the rule for resolving a fraction into partial fractions 4,(x,...x„) _^ ia„a,... «,) /'K)/'K) •••y'(a,)(^.-«<)(^.-«.) ••■ K-«.)' ,.(2). 16, 17.] EATIONAL FUNCTIONAL DETERMINANTS. 127 Now, in the first place, ia the combination i,k...p, no repetition . can occur, for in the product not only B, but also ^•' ^ ^'' vanishes if x, and x„ both coincide with a^. Hence on the right of (2) we must write for i,k ... p all permutations of 1, 2 ... n. Now if we write a^, a^ ... a^, for x^, x^ ... x„ respectively, only a single term of (Wj ... uJ^B remains, viz. + [/'(«<),/'(«.) -/K)?, while ^i{x^,x,...x„)=^i{(X„a,... a^) = ±^(a^, a^ ••• a„). the ambiguous sign being the same for both. Thus = (-1)'^/' («.)/' («,).../'(«,). Thus where i, it ... ^ is to be a permutation of 1, 2 ... n. This proves the theorem as stated at the beginning. 17. The coefficients in the expansion of the rational fraction 1 +\x + b,x' + ... l + a^x + a^x''+ ... ' in ascending powers of x can be represented as determinants. Viz. if the expansion is 1+P,x + P,x' + ... we have {1+b^x + b,x' + ...)=' {I +P,^ + P,x''+...)i1 + a,x + a,a^ +...), 128 THEORY OF DETERMINANTS. [chap. IX. and hence equating coefficients a^^P^ + a^P^ + . . . + P„= K - a„, a system of equations to find P„. The determinant of the system is unity. Hence, if after solving by viii. 1 we move the last column to the first place, and change the sign of this column Pn = {-IT a^-b^, a„ 1 =(-ir dn-K, Cf«_l. «^2 ••• • ■•. 1, 1, ' > • k, a„ 1, • . \> «2. a„ 1 • 9 K> a„ a„ a. 1 as we see by subtracting the .first .column from the second in the latter determinant. CHAPTER X. ON JACOBIANS AND HESSIANS. 1. If y,. 3/3 ••• 2/„ be 11 functions of the n independent va- riables a-^, .Tj ... a;,,, and if then the determinant | a^ | is called the Jacobian of the functions 2/j ... y„ with respect to the variables .i\ ... x^. The name was given by Prof. Sylvester after Jacobi, who first studied these functions. The notations have been employed for Jacobians, each of which has its advan- tages. The first renders evident the remarkable analogy between Jacobians and ordinary differential coefficients. The second is useful when there is no doubt as to the independent valuables. If the y's are explicit functions, the Jacobian is formed by direct differentiation. 2. If the functions y^ - - . y„ ai^e not independent, but are con- nected by aa equation ^(y.. y, •••2/J=0. the Jacobian vanishes. For if we differentiate this equation with respect to x^, we get dy, dx, di/, dd', ' di/„ dr, ' s. D. 9 130 THEORY OF DETERMINANTS. [chap. X. where ^ = 1, 2 ... n. Eliminating d(f> d d(p dy^' ^2 """ dVn from these equations we get (vill. 3) d{y„ .V, ••• yJ _Q 3. If the functions y are fractions with the same denominator, so that u. 2/^ = 77' jdy, _ du. du dx^ dx^ 'dx. Thus dK ... x„) u, «1> 0, ^ cZm, _ du dx„ "' c^a;„ c?w„ du dit du M„, W C^i», "rfa;. u-~ — u„^r- dx„ dx. du Add the first column multiplied by t— to the (i+ 1)" column, (tsc. and we get i t^(yi ••• yJ M, M du dx^ du. whence dividing each of the last n columns by u u, du du dx^ dx„ du^ du^ dx^ dx„ du. du^ dx^ dx„ 2—6.] ON JACOBIANS AND HESSIANS. 131 4. The determinant on the right has been denoted by K{u, Mj ... u^. It has interesting properties of its own. For example, since the Jacobian vanishes if the quantities y^ ... y„ are related bj an equation, it follows that K [u, M, ... mJ = if a homogeneous relation exists between u, u^ ... m„. If it is readily shewn that t' K{u, Wj...wJ = ^^(v, v^ ... v„). 5. If the functions J/i •■■ y^ possess a common factor, so that ^ #2 dx^' dx^ 0. 0, -f^... dx. 1, 0, ... '^'^^ 1 d{y.. d{x^ . -Vn) It follows then that diy, ... y„) _ #, c?<^, dn. d{x^...xj But if m < n ^(^1 ••■^J _-< d{z„ z^... gj d{y,,y^,y,...) d{oo,... xj d {y„ y„, y^ ...) ' d {x^, x^ ...xj ' where for t,u,v ... we take all m-ads in n (iv. 2). 8. If _^ . . . /„ are independent functions of x^ ... x^, then iTj ... «„ are independent functions of/j .../„, and we have d {./,■■■/„) d{x^...x„) ^ d{x,...xj-d {/,.../„) '■ For differentiating y|. with respect to/^ we must consider a;^ ... a;„ to be functions of /j .../„. Thus Mi ^^§fi ^2^ _^§fi dx^ da-\' df^ dx^' df^ ■■■ dx,,' df^ is equal to unity or zero, according as h is or is not equal to i. Hence dx, dA dx. dA = 1. For in the product only the elements in the leading diagonal do not vanish, and these are all equal to unity. dx. 9. If J. = and J.,j, B.j^ are the complements of JB = df, dx. dL minants we have , and -nr , in these two deter- dx^ df^ A Jy; - A.. ^1 = ^- 7 — 10.] ON JACOBIANS AND HESSIANS. 135 Also . cZ(a;,...a;J _ d(/„^,.../J For we have just seen that ^ dx^ df^ dx^ df^ dx,,_ dx/df,-^dx,- dfj-+dx„-dfr ^ dx^,^ d:>\, ,df^ dx^n^-, d (^2^2 ••• d^^n (1)> dn^i, d„x^ ■■■d„x„ let the corresponding increments of the functions be d,A, dj, ... dj^ dj„ dj^ ... dj„ (2). dnfv dj^... dj^ Then just as the differential coefficient of a single function of a single variable is defined to be the limiting ratio of corresponding incre- 10 — 12.] ON JACOBIANS AND HESSIANS. 137 ments of the function and variable; the Jacobian of the functions f^ ...fn of the n variables x^ ... «„ is defined to be the^limiting ratio of the determinants of the systems of increments (2) and (1). That this leads to the same Jacobian as before is plain from the equation which gives (iv. 3) \dJA__ d{f,...f„) \d^x.\ d{x^...xy according to our former definition. Using this new definition we can prove all our former theorems. Let us use it to prove the first of the above equations, viz. the theorem of Art. 7. If the system of increments given to a;, ...«„ be d^x^ ... d^x^ d^x^... d„x^, let the corresponding systems for y^..- y^ and z^... z„ be d^y^—d^y„ d^z^...d^z. "I'^/i dnVx-'-d^y^ d„s^...d„z^. Then we have identically I d,z^ I _ I d,z, I 1_^|. I d,x^ I I ^i2/J ■ I d^x^ I ' or by definition, d{z^... 2 J ^ d{z^ ... z„) d{y^...y,^ d{x^ ... xj d{y,... yj ' d(x^... x„) ' 12. We can also, using alternate numbers, obtain a symbolic expression for the Jacobian, from which the ordinary results follow. Viz., y^...y„, being n functions of a;^ . . . «„, let X = e^x^ + e^x^ + . . . + e„a;„. 138 THEORY OF DETERMINANTS. [chap. X. Then dx, ^dx,^^dx^^---^^''dx,' 3nce (i. 19) dy dy dy _ dx^ ' dx^'" dx„ ^i_ dy, dx^'" dx„ dyn dy„ dx/" dx„ _d{y,...y„) d{oc^-..so„) But now dy dy dx dx^ dx ' dx^ ~^'dx- Thus the above equation (1) becomes fdyY^d(y^...y„) (1). \dxj d{x^... xj From which symbolical equation we can deduce our former theorems. For example the equation \dx) \dy) gives at once d(y,...yj d(x,...x„) _^ d{x^...xj ■ d(y^...yj 1.3. Jacobians occur in changing the variables in a multiple definite integral. Let us transform the integral I=jj...F{y^...y„)dy,...dy„ to an integral with respect to a;j . . . x^, the functions y, ■ • • 2/„ being supposed given functions of «, ... «„. We proceed in the manner used by Lagrange to transform a triple integral. Beginning with y„ we have to find the sum of the quantities ^dy,^, 12, 13.] ON JACOBIANS AND HESSIANS. 139 while y^,y^... y^^ remain constant. This gives us Solving this to find dx„ we get (vili. 1) u («J -dx. where Thus J a. 'J a, J a. ^i («i) 02 K) • •■ ^„ W (exp. M = e"). Now let us introduce in place oi os^, x^ . . . x^ the n new variables Vi ■■■ Vn' gi^^GH ^y the equations Vi 1 y^ , 1 Vn _ j^ Then by ix. 13, and hence da;, a;, — tti Thus by ix. 11, d{x^...xj y.-'-Vn aj, — a. = (-1) ?^i^ y,-yA^ K ••' ^J ^^ K ••• ^J 71 (71+1) Now 142 THEORY OF DETERMINANTS. [CHAP. X. Hence in the integral we replace dx^... dx^l^^ (x^... x^ by dyx--dy,^{a,...a^). Now if we write F„, («) = (2 - aj . . . (^ - a J {a^,- z) ... (a„ - z) we have 2/1^' y-. P.„P2 V^-- '^l(^l)^^K')---'^nfc.) Hence is replaced by dy^...dy^ i,^{a^...a^) y^'...yj"'F,'{ay^...F,:{ajP''- Again x^ ... x^ can be regarded as the roots of the equation Vl ^ 3/2 J. ___ J_ Vn -. ]^ e — a^ s—a^ '" z — a^ ' the roots of which lie between a^ and a^; a^ and a^... a„ and 00 . Hence y^ ... y^ take all positive real values. Also we have x^ + x^+ ... +a;^ = y^+y^+ ... +2/„ + a,+ ... +a„. And our integral reduces to (- 1)' ° ^^ K •■• O exp. (- g^ - ... - gj i^;(aj^'...i?;'(a„)2'» r exp.(-y.---yj _ Jo yr-y^" ' 2/1^ (- 1)"'"-"- r.(i - pQ r (1 -pj ... r (1 -;,j ^_,._„._ _ {i'/(gj^^-i...i^;(gj?'»-i}* 15. If M be a function of n variables x^,x^...x„ and y^ ... y„, its differential coefficients with respect to these variables, since ^Vi _ ^ (du\ _ cZ^M c^a;^ dajj \dxj dx^.dx^ 14—16.] ON JACOBIANS AND HESSIANS. 143 Tlie Jacobian o{ y^ . . . 2/„ is a symmetrical determinant formed from the second differential coefiScients of u. This determinant is called the Hessian of u after Hesse, and is denoted by H (u), so that H{u)=\u,\. The Hessian of u will vanish if the first differential coefficients of u are not independent (Art. 2). For example, if u = / + . . . + <_,^„^ dx,dx~ * *' •. H{u) = 2« + <+...+0> ix^w^ ^SC^OCn ,2« + <+...+0- Or dividing the i'^ row by 2x., and the A;"" column by 2^^ H{u)={Tx,x^...xy <' + X + <■ 2x; 1 2x: This is a determinant of the form of that in III. 25. If we write V = {a-xl){a-x:)...{a-x:) If u = xY + fz'+z^af, this gives H (u) = 24 {dx'yW - {a? ->rf + a') u]. 16. Jacobians and Hessians belong to that class of functions known as covariants. That is to say, if these functions are trans- 144 THEORY OF DETEEMINiSTS. [CHAP. X. formed by means of a linear substitution, the Jacobian of the transformed functions is equal to the Jacobian of the original function multiplied by the modulus of the substitution, and the Hessian of the transformed function equal to that of the original function multiplied by the square of the modulus. Namely, if the variables be transformed by the substitution ^i = ot.ii + «.2?2 + • • • + ajn (*■ = 1, 2 . . . n), the determinant | a^. | is called the determinant, or modulus, of the transformation. If the functions y^ ... y^ oi x^ ... x^ when transformed by this substitution become the functions y/, «// ... y\ of ^^ ... ^„. Since dyl^dy^da^_^dy^dx^_^ ^dy^ clr^ d^, dx^ d^, dx^ d^, •■■ dx„ d^^ du. dy. it follows from the multiplication theorem that d(y,' ■■■yj ^ d( y^ ... y„) . d{^,...L) d{x,...x„)^''--'\' which proves the theorem for Jacobians. The theorem for Hessians follows from this, viz. if m be the original and u the transformed function. Since the Hessian of u is the Jacobian of ^ — . . . —. — we have dx^ dx„ J fdu du du'\ d (du du'\ [dKriMl Now d K .-; «„) d^u d^u dx^d^t d^^dx.' J fdu du\ \dx, '" dxj IG, 17.] ON JACOBIANS AND HESSIANS. l+c 1 [du ' dn du ^ dx 17. If we have n linear functions 2/, = ^.+ ••• +^a(^ = 1, 2... «)■ If M is a quadric function "■ = ^11*1' + • • • + '^I'J-x,, + . . . , then ^(«) = 2"|6,, |,C&. = 5,). The symmetrical determinant on the rights which is called the discriminant of the quadric, is therefore an invariant which on transformation is multiplied by the square of the modulus. S. D. 10 CHAPTER XI, APPLICATIONS TO QUADEICS. 1. The general quadric function in n variables x^.,. x^ is denoted by the coefficient of x^ being a^, that of 2x^x^, a.^, and we suppose By X. 17 the symmetrical determinant A = | a^ | is propor- tional to the Hessian of u, and is hence an invariant, it is called the discriminant. On transformation it is multiplied by the square of the modulus of transformation. Let us write 1 du = a,,x^ + a,^x^ + ...-^a,^x„. 2. If we form a new quadric whose coefficients are the com- plements of a,j in A, viz. IT is called the reciprocal of the given quadric. We may also write it in the form (ill. 25) U=- 0. 2/l ■■■2/n 2/.. o^u ■••«,„ 1—1] APPLICATIONS TO QTJADRICS. 147 Since | A^ | = ^1*"', and if a^. is the complement of A^ in this detemiinaut a , =n,j.-l"~* (v. G\ we see that we can write u in the form A'-'u = - 0, .r. . •• •'•» ''■i. Ax ••A- •'",.. -Ki •■-i:.., "We have also ^M=- 0, V, .. w« "„ «., .. ".,. 1 ".. «-f • 0^ as we see bv multiplying the last n rows by a\ . . . a:„ and subtract- ing their sum from the first. 3. If J. = 0, since then ^\ = A,A. it follows that is a complete square, and that the lineo-linear function V=- 0. .r, ... X, . A... is the product of two linear factors. •i. The reciprocal quadric U is the first of a series of co- voiiant quantics. If the variables .i\... x^ are transformed by a linear substitution «*=f",i-'-i' + c,i-»V+ ••• + 'V.'"-.' W (* = !.- ••• «). 10—2 148 THEORY OF DETERMINANTS. [chap. XI. then the function becomes •■•+< ("1,2/1 + 02,2/2 + ••■ + c„<2/J + ... Hence, if we have a series of quantities y^ ... yl given by 2// = Ci,2/i + C2,2/2+...+c„,y„ (2) (i=l, 2...n), the function %x/y. on transformation becomes changed to "^xlyl, and so is absolutely unchanged in form by the transformation. Now observe that in the substitutions (1) and (2) the deter- minants of the transformation are identical; only the columns of the determinant of (2) coincide with the rows of the determinant of (1). Also in (1) the old variables are given in terms of the new, in (2) the new variables are given in terms of the old. The variables ajj . . . ij;„, 2/1 ■ . • 2/„ are said to be contragredient. Any function of the coefficients of u and the quantities y-^.-.y^ whose value on transformation is equal to its original value multiplied by a power of the modulus of transformation is called a contra- variant. The semi-differential coefficients m, . . . u„ are contragredient to 5. If the p sets of n variables 2/1: ••• y.i 2/12 ••• 2/.2 2/ip ■•• 2/"jj are contragredient to the variables x^...x^, then the series of determinants R.= «11 ••• «!„. 2/11 ••• Vlp ^nl ■■■ ^nn) y„i ■■■ Vnp 2/„ •■• 2/« 2/lp ••• Vn are contravariants. 4—6.] APPLICATIONS TO QUADEICS. 149 For, let us consider the quadric function F= ta,,x,x^ + 2i, {x^y^^ + . . . + x^yj + 2«, (a^i2/i2 + • • • + XnV^ + • • • + 2^, (x.yi, + . . . + x„yj, where the variables x^...x^,t^...t^ are cogredient (i.e. transformed by the same substitution), while y^. are contragredient to these. If we regard F as a quadric in-n + ^ variables x^...x„,t^...t^, Rp is its discriminant. Let us transform it by means of the sub- stitutions x, = c,X + ---+c,,<] t, = t: k = l,2...p. (i=l, 2... n) Then the determinant of the transformation for x, t is u = Cn • • c,„, . . . •■ c„„, . . 0, 1 . . . 0, ... 1 = c,. In the transformed function V, the terms multiplying i, are unaltered in form. Hence, by Art. 1, Thus R^ is a contravariant. Since on transformation it is multiplied by the square of the modulus, it does not change its sign. 6. If j9 = 1 so that we have only one system of y's, then R^ is the reciprocal quadric. If for uniformity we denote the discri- minant by R^, we have (in. 25) .^ dRij da„ And in general we have dR^ ' da,. -^"■JJ+l ^ rlr, Vi. -P+l y^' *+!■ 150 Clearly THEORY OF DETERMINANTS. [chap. XI. i?„=(-ir 2/u ••• y,, Vnl ' ' ' Jn'' while JBj, vanishes identically if p is greater than n, as we see by resolving it into the sum of products of complementary minors of order n and p. Thus we have the series of functions containing 0, 1, 2 ... n series of variables y, and of orders «, w — 1 ... 1, in the coefficients of the quadric u. 7. The determinants R^ are of great importance in the dis- cussion of the properties of a quadric, and especially in the reso- lution of the quadric into the sum of squares of functions linear in the variables x^ ... x„. If Mj ... M„ are the semi-differential coefficients of u, let us write u,- «u • ■ «!.. 2/ll • •• Vip' «! dra- • a„», 2/„i • •Vnp, w„ 2/n • • ym 2/ip- ■' "np «, . ■■ W„ We must remember that C^„ = identically. Also let X^ be the determinant obtained from U^ by erasing the (w -h J3 + 1)" column and (w ■\-pY°' row ; or the (w +^ -I- 1)" row and in -Vpf" column. Since in any determinant of order m, we have (v. 7) d^B dD dD dD dD D da^... da da^ 1 „, , ' da^ 771—1 ni— 1 m <^'^m-lm ' -l ' we get by applying this to U^, (m=n+p) or ^f ^p-i - ^p = ^t-i ^' R«. 6—8.] APPLICATIONS TO QUADEICS. 151 In this equation write p = n, n-1 ... 1, 0, and remembering we get the series of equations U X Thus M = — «> Now the quantities A\ ... X„ are linear functions of m^ . . . m, i.e. of .fj ... d'„; hence we have resolved the given quadi-ic into the sum of the squares of 7i linear functions of the variables x^ ... ^r,,. Also the number of positive squares in this sum is the number of variations in sign in the series -^0. -Rj ••• R.„ and these being unaltered in sign by a linear transformation we have the important theorem, that if a quadric be hnearly trans- formed to the sum of ii squares, the number of positive and negative squares is always the same. This theorem, due to Sylvester, has been called by bim the law of inertia of quadratic forms. 8. The discussion of the preceding article, due to Darboux, requires modification in certain cases. For example, if the minors of order ^ — 1 of the discriminant vanish, then all the functions i2g . . . i?p_j inclusive vanish. In this case Darboux has shewn that M can be resolved into the sum oi n—p squares, viz. -Rj+l-^J- " -K„-i-K«-j Rn-l^« 152 THEORY OF DETEEMINANTS. [CHAP. XI. 9. If a quadric, by means of a linear transformation, has been reduced to the sum of n squares, u = tajc-x^ the discriminant of the right-hand side being A^A^ ... A„\{ fi is the modulus of transformation, A^A^...A, = IJ? I aj. Two given quadrics can by a simultaneous linear transformation «'i = Ca2/i + c«2/2 + ••• +c*»3'n (^ = 1, 2 ... n) be reduced, each to the sum of n squares of the same linear func- tions, viz. V = s.A^y^ + s^A^y.^ +... + s^A^y^; for in order to determine the rf constants, c.^, we have first n {n — 1) equations from the fact that the coefiScients of the products y.y,^ must vanish, and n additional equations from the condition that the ratio of the coefficients of y^ is to be s., in all n^ equations. If we form the discriminant of su — v, its value for the original quadrics is I «a*-^« I (1)' and for the transformed quadrics A,...AAs-s,){s-s,)...{s-s:) (2). The ratio of the quantities (1) and (2) is fi^; hence s^ ... s„ are the roots of the equation A(s)=|sa,-&J=0 (3). _ 10. The following resolution is due to Darboux. If we write F=su-v, X = i -J- =su,-v. (4), ax. 9, 10.] APPLICATIONS TO QUADRICS. we have identically by Art. 2 153 F= su — v = — A(s) «C^n - ^1 ««ln-^„. ^1 ,(5). Xj ... z„ The determinant on the right is a function of s of order n — \; resolve the fraction into partial fractions, and we get 1 SM — « = — 2 A'(s,)(s-s,) «;«„!- 6„i a: X. ...(G). The determinants on the right are all perfect squares by Art. 3, for they are obtained by bordering the vanishing determi- nant A (Sj). Whence su-v = Z .,, . / -^ , where U^ is a linear function of the form If in the determinant (6) we replace X. by its value from (4), and subtract from the last column the first n multiplied by «j ... «„, and do the same for the rows, the value of the determinant is unaltered, but X. is replaced by |- (s — s) -y- . A term is also introduced in the principal diagonal in the last place, but since its minor vanishes by (3) we may replace it by zero. Thus U. is replaced by TT/ 1 / s f T du , du where V. is independent of s ; .-. SU—V = 2t \\ / s ■ Equating coefficients of s we get « = ^a1^ which is the required resolution A'(«.) 154 THEORY OF DETERMINANTS. [CHAP. Xt. 11. An important branch of the theory of quadrics is that of their linear automorphic transformation. That is to say, as the name implies, the discussion of those linear transformations which do not alter the outward appearance of the quadric. So that if «j ... a;„ are the original, and y, ... y„ the new variables, ta^x,x^ becomes ta^,y^. Without entering into a discussion of the general case we shall study that particular one which gave rise to the whole theory. In the transformation from one set of rectangular axes in space to another with the same origin, the distance of a point from the origin is the same, expressing this for the two systems such a transformation is linear and automorphic, and is known as an orthogonal transformation. 12. The general case of an orthogonal transformation is to determine those linear transformations which give us < + <+... +a;„^= 2// + z//+..-.+.y„\ The theory is due to Cayley, but we shall here give it as modi- fied by Veltmann. Let us consider the following equations ^11*1 + K^^ + ■ • • + &iA = \,y, + \,y^ + --- + b^„ ^1^1 + K^i + ■■•+ &,A = b,,y, + b^^^ + ... + b^„ &„i«, + &„2*2 + • • • + &„ A = 6i„2/, + b.^y^ +...+ b„^„ where the system 6.^ is skew, so that K = -Ki, h = z (2). The rows of coefficients on the right coincide with the columns on the left. ' Let B=\h,,\ = \b,,\, so that 5 is a skew determinant, let B^^ be the system of first minors. Solving the system of equations (1) we get 11 — 13.] APPLICATIONS TO QUADRICS. 155 The coefficient of x^ in ?/, is given by Be,, = 5A + -5,A* + • • • + BJ^. Now s = 5 or according as i is or is not equal to k, thus 2B,,z 2B,,z-B In the same way ^ _25,.« , _ 2B,,z-B Thus c^=d,„ and we may write Substitute for 3;^ . . . «„ from the second of these systems in the first and equate coefficients of y^ and y, on both sides, thus Ca^a + CfflC,, + . . . + C,„C,„= Oj ' If we substitute from the first system in the second, we get cj + cj+... +cj = l) Whence we see at once that < + <+... +x: = y,' + ,j,'+ ... +y:, and thus the coefficients c,^ are those of an orthogonal sub- stitution. 13. By the preceding article we are able to express the n' coefficients of an orthogonal transformation by means of the J n (ji — 1) quantities 6„. ... b^ by forming a skew determinant with these, the elements of whose leading diagonal are equal to z. 156 THEORY OF DETERMINANTS. [chap. XI. For the case n = 2, let B = the system of first minors is 1, X I = 1 + V ; -X, ll 1, X X, 1. Hence the coefficients of a binary orthogonal transformation are 1-X^ 2X 1+V l+X^' - 2X 1 - X' 1+X^' l + V For a ternary orthogonal transformation B = = l+\' + ,j,'+v'; 1, V, - fl — V, 1, X /A, - X, 1 the system of first minors is 1 + X^ V + X/i, - ij, + Xv, — v + Xylt, 1 + /^^ X + fiv, fi + Xv, — X + fiv, 1 + v''. Hence the coefficients of the ternary orthogonal transformation are l+X'-zx^-i^' B ^ B ' ^'~B^' B -X + /XV X + /«/ ^ 'B~- l + i^'-X'-z^" B If we write X= cos/tan ^^, /u. = cos ^^ tan ^^, y = cos A tan ^^, where cos^/+ cos^ g + cos'' A = 1, and we get Rodrigues' formulae. 5=sec'-i^, 13 — 15.] APPLICATIONS TO QUADRICS. For the quaternary orthogonal transformation B = 157 1, a, b, c - a, 1, K-g -b, -h, 1 / - c, 9, -/- 1 Then B = \+a? + ¥ + c'+p + g'' + h' + e\ where 6 = af+ bg + ch. And the system of first minors is Aa= l+r+/^-^^ B,, = -a-fe + cg-lh, B,, = -b-cf-gd^ah, B^, = -c + bf-ag-he, -^13 = b+gd-cf + ah, ^.3= h+fg + c9-ab, ^33= l + /+c^+a^ ■^43 = -f + gh-bG- a9, B^^= a+fd-bh+cg, B,,= l+f'+b^+c-', B^,^-h +fg -ab- c0, ■^42 = g +/^ + be- ca, B^,= c+he-ag+bf, B,, = -g+hf-ac-be, B^= f+gh+a0-bc, B = l + h'+a'+b'. Thus the coefficients of the quaternary orthogonal transforma- tion are Be,, = 1 - 6' +f - a' + g' -b' + h' - c\ Bc,, = 2{a+fe-bh-[-cg), Bc,, = 2{b + gd-cf+ah), Bc,, = 2 (c + he-ag+bf), &c. 14. The square of the determinant of an orthogonal substitu- tion is unity, for where i.e. I c or I c,,| = where e means ± 1. 15. If C;j is the complement of c.^ in C, then C =e.c„... 158 THEOKY OF DETEEMINANTS. [chap. XI. For we have the system of equations c,jCj^+... +c„iC„, = ^Vc'^U I" • ■ • + ^ni^rt — i Multiply these equations by G^, G.^... C,^ and add, the co- efficient of o^^ is e, the others vanish, thus 16. Any minor of the system c^^ is equal to its complementary minor. For by V. 7. But G„ ... C,, o^. ... o«« Gn- ■0., = 6^ Cll • •c,. On- •c,. C^i- •«i>i> by the theorem just proved. Hence 17. If ^"" = i ttft I -B'"' = I 6^ I be two determinants of ortho- gonal substitutions of order n, then the determinant P (\ fi) =-\\a^ + A, I is not altered by interchanging \ and fi. For the symbolical expression for P (X, (i) is P{X,^) = {XA+fiBy l(,.)R(re|/^ . '^^ as in v. 8. And as there proved 15 — 18.] APPLICATIONS TO QUADRICS. loi) Or, if .4"" = 1 = JS"", we have by Art. 15, P(\, /i)= I Xb^ + fia,, I From this we see, that if from the coefficients of an orthogonal substitution of order n we subtract the corresponding coefficients of another orthogonal substitution of the same order, the determinant formed with these differences vanishes if n is odd. 18. If we take n quadrics in n variables we may conveniently represent them by the system of equations «i = Sffly^a-jaj^t (.i j, /i; = 1, 2 . . . «)• With the coefficients a^ we can form a cubic determinant of order n which will be an invariant of the system of quadrics u^ ■■■u„. Zehfuss has pointed out that for three ternary quadrics this gives Aronhold's invariant, while the auxiliary expressions he gives for its calculation are the cubic minors of the second order. For the two binary quadrics aj: + 2hxy + cif, a\v'+ 2b'.ri/+ c'y", it is the harmonic invariant aa' — 2bb' + cc. The general theorem is that for n, n-ary jj""' the determinant of class (p + 1), which can be formed with their coefficients, is an invariant of the system. By allowing all the quantics to become identical we get an invariant of a single quantic when it is of even order. CHAPTER XII. DETERMINANTS OF FUNCTIONS OF THE SAME VARIABLE. 1. If y^, 2/2 ■ • • Vn ^'^s functions of a variable x, and if yr the determinant S + y.2/,"'. ■ y'r = y, > y^ ■ ■■ 2/,. ■■ 2/J" in-V in-l) y\ > y^ .. 2/r'- is called the determinant of the functions ^i, 2/2 ■ • • y„, and is denoted by -D (2/1. y, ••• 2/J- 2. If 2/ is any function of x, and we multiply the above deter- minant by y • ... /', y ... y'\ 2i/''' , y ... y' ,11-1) 1)^2/'"-', («-!),/ »/ = 2/ combining the columns of B with the rows of the latter, we obtain ^ toi. 2/2/2 ••• yyn) = y"D{y„ y^... yj. In particular if we put yy^ = 1 in the determinant on the left, all the elements in the first column vanish, except the first, which 1 — 3.] FUNCTIONS OF THE SAME VARIABLE. 161 is unity, and the determinant reduces to the determinant of the n — 1 functions doo\yJ 2// '" dx\yj y^ If therefore we put ^ (3/.. 2/.) =2// ••■ ^ (y„ y^ = y„'. then D (y^, y^... 2/ J = -v-. -D {yl, yi ... yj. 3. If the functions y^ ... y„ are connected by any linear relation it is plain by differentiating this n—\ times, and eliminating c, ... c„ between the original and these n—\ new equations that we get : ■0(2/1. y^ •••3/n) = 0- Conversely if the determinant of the functions y^ ... y^ vanishes, then they are connected by a linear equation with constant co- efficients. We shall prove this by induction ; we shall assume that if the determinant of w — 1 functions vanishes, these functions are linearly connected, and we shall shew that the same is true for n functions. If y, does not vanish, which would be equivalent to a linear relation among the functions, it follows from the pre- ceding article that since ■0(^1, y2---2/J=o, . we must also have I>(^i 2/B'-2/n') = 0. Hence by hypothesis the n—\ functions y/ ... y„' are linearly connected, i.e. we have 0,2// + '-8^3'+ ••• +<'„2/„' = 0. Dividing by y^ we get '^Tx\yJ^''dx\yJ^-^'"dx\yJ *"' or integrating Ci2/i + c,2/.+ ■•• +c„y„ = 0. S.D. 11 162 THEORY OF DETERMINANTS. [CHAP. XII. Thus if the theorem is true for n — 1 functions, it is true for n, but it is clearly true for two functions, and hence generally. 4. From the formula -D (2/1, 2/, ... yj =—^D{y.l, 2/3' ••• Vn), «7l it follows that Divv 2/2. ys) = -^iy,'> .Va') -0(2/i> y^. 2/4) = --0(2//, 2/,') 1 -D(2/i. y,> yn)=-D(y^, yj. ji The same formula also gives ■D{y:, y; ... 2/;) = i^{^(2/;. 2/3'). i)(2/;> 2//) ... ^(2/;> 2/;)i- Combining these formulae, we obtain the equation -0(2/.. y, •■•• y"^^ [j(y^, y)]"-3 -0{-P(yi, y„ y,), D{y„ y^, yj ■•■ ^(y.. y2. yJ}- By repeated application of this method we should obtain the theorem. If Mj, u^ ... u^, V,, v,^ ... v„ be functions of a;, and if Wi = D{i\, u^ ... u^, V,) (i=l, 2 ... n), then D(u u u v v v)- ^ ^'"'' ""-■-'^^ then i^ (M„ M, . . . «„, ^1' ^2 ■■•"»-'-{ /^ (m^, M^ .. . M^^4-» • 5. A special case of this theorem is -0(y, ... y^-i, y,+i ... y„, y,, y) _ I){D{y,... y,_i, y,^, ...;/„, yj Z)'(y, ... y,_i, y,^i ... y.„,y)] -0(yi ...y*-!, y^+i ... yj which we may write in the form ■P(.yi ... yn, y)-P(yi ... y^-i. y.^-i •••yn) ^ d D(y,y,...y,^i,y^:,^...y„) •0(yi •••yn)^(y, ... y„) <^.c i){y^...y„) 3—6.] FUNCTIONS OF THE SAME VARIABLE. 163 Assuming now that the functions y^ ■•■ yn are independent, let us write ^.= (-1)" D{y.-y.) then the above equation can be written 6. The determinant y, > y, yr> 2/.'" yn y. (1) l»-2) (B-2) y'^' , y? y: 2/J* vanishes \ih. + •••+««'"■>„= ■(B). ^r\ + ^r\ +■■■+ c-''y. = (- 1)"" li JJ (z^, z^ ... z„) = vanished, it would follow that since Soo = 0) *oi ~ ^ • • • ®cta-2 — 0, then S|^_j would also vanish, while its value is (— 1)""'. Thus the functions s,, z^ ... z„ are. not linearly connected with each other. Comparing the systems (A) and (B) it appears that the relation between y^ ..■ y„ and z^ ... z„ is a reciprocal one, if we neglect the sign when n is even. From each relation between these systems we deduce a new one by interchanging y,. y^ ••■ I/n, ^.> «2 ••■ ^» with (- 1)-'^., ( - ir\ ■■■{- ir\. y., 2/. ■ • ■ i/n- Thus from the equation ' ^ ^ ^{y..y.-yny we deduce y" ^ '^ D{z„z,...z„) ■ In consequence of this we shall call z^ ... z^ the conjugates of y,--y.- 6—8.] FUNCTIONS OF THE SAME VARIABLE. 165 8. ]f we form the product by rows of the two following deter- minants y, ■ •• y,> 2/*+. • • yn yr' ■ ■yt^ . . yr' y^ ■ yW •• yt > yf.^ ■ ■ yJ" yr" ■ •• y* ' ■yr' 1 .. 0, ■ ... .. 1, ... ^1 ■• ^*. ^M-1 ... z„ , (n-Jr-1) (n-i-1) ln-*-l) ■• z. (m-S-l) the first of which is I>{y^... jrj, the second B (a^j ... z^ we get y, • ■■ y.. «oo ■• • 5o»-j-i yr" ■ y^" ■ •• yr'> ■ yr, yr'- ■ yr\ ^n-10- • ^n-ln-*-l In this determinant the block of elements common to the first h rows and last n — h columns all vanish, whence it reduces to S'l yr-y y^ °»-10 • • ■ °ii-ln-4-l The first of these =D{y^...y^, in the second all the- elements to the left of the second diagonal vanish, whence its value is (»-M(n-fc+l) \~^) *»-io *»!-2i • • • ^m-i-i = 1. Thus we have If A; = we have D{y,...y:)D[z^...z„) = l. 166 THEORY QF DETERMINANTS. [chap. XII. 9. From this last equation we get =^{-rrD{y,y,...y:iD{z^,z,...z„). Or P{y) = {-Vr = (-ir y> Vx ••• Vn y'''\yr-yT' y > *nO>®»l ••• ^nn-1 1, 0, z. 0, z^" Similarly we should get ^oo> *oi ••• *0i *„_10> S„_ii ••■ S„_,„ ..z. (n-l) 10. These determinants occur in the theory of linear differ- ential equations. Thus, if we have the equation ,1") — 1 where the quantities a„, a^ ... a„ do not contain y. Then iiy^...y„ are n particular integrals, we have the n equations ao2/<+'»,yi'"+--- + «»2/r = o eliminating the tx's we get y, y'" 2/i. yi' (i = l,2...n), 2/"" =0, ■ y^' 2/.>2/„'"--yJ"' D[y,y,-..y,) = 0. 9 — 11.] FUNCTIONS OF THE SAME VAHIABLE. 167 If we solve the equations for -^ we get 2/x.3/,'"-.vr",2/r y.„y:"--yr'\yr Vvvr-yr' y,.-2/,.'"-3'.r' a.. I.e. logD(2/,,y,...2/„) = - -^\ dx 11. Though not immediately connected with the subject of the present chapter we shall give Hesse's solution of Jacobi's differential equation. This equation is where A. = a.^ ^ + a.^-q + a.^ (t = 1,2, 3). We can write the equation in the form of the determinant I '?, 1 =0. d^, dr), Now let ^ = - ; v= > ^^^ the equation becomes 0. 00, y, z zx — z'x, zy — yz', A^, A^, A^ Multiply the first row by / and add it to the second, this divides by z, and we get X, y, s =0. / / r X, y, z Now let us multiply this equation by "2. ^2' 72 , «3> ^3' 73 ! and let p, = ol.x + /S.y + 7,^. 168 THEORY OF DETERMINANTS. Also assume that [chap. XII. ^iPi = A«, + A0i+Ayr Then Pi' P^> A ^32' ^33 "'^ = 0. CHAPTEK XIII. APPLICATIONS TO THE THEORY OF CONTINUED FRACTIONS. 1. The application of the theory of determinants to continued fractions is one of its latest developments, and gives great facility in the discussion of these functions. As usual in English mathematical works we shall denote the continued fraction a™ by _L_A A_ An «1+ (^^+ «8+ +»« Such a fraction is called a descending continued fraction. In addition to these we shall discuss a less known form of continued fractions, which, however, is historically the older form of the two, namely, the ascending continued fraction which, in an analogous manner, will be denoted by a, a, ■■■ a„" Our object is to establish a determinant expression for the convergents to these two forms, 170 THEORY OF DETERMINANTS, [CHAP. XIII. 2. If we write down the system of equations h^x = a^x^ + «j ^2^1 = "'A + ^3 h<", = «3«3 + ^4 we see that ^ &, 3-, 6, X X, X^ , «, «i + a^ + — Hence -^ is the continued fraction X 3. If we are to determine the w'" convergent, i.e. the value of the fraction when we stop at — , we must suppose that x'^-^ and all succeeding x's vanish, whence we have the system of equations h^x = a^x^ + x^ = - \x^ + a^x^ + x^ = - \x^ 4- a^x^ + x^ 0= -&A-i + aA. Solving this set of equations for x^ we get : Thus a, , 1,0.. «'i = = b^x, 1,0. -h. a^ , 1 .. 0, a, , 1 . ,- -63, a,.. a„-l , 1 0, -&3. «8- , 0,0.. 0, 0,0. ■■-K> <^n , 0.0 ... -K, a,n a; \ ^2 > 1 -^3. ^8 ..0,0 ... 0,0 ■-7- «1 . 1,0.. a^ , I .. -^'3. «'3-- .0,0 .0,0 .0,0 , , ••«„-x. 1 .. -b„, a„ • u , , 0,0.. • a„-i, 1 , 0,0.. • —K> £*« 2—4.] Or THEORY OF CONTINUED FRACTIONS. 171 -' =-^ say. X Where Pn=K a, , 1 , 0, ... , -h„ a, , 1, 0... 0,0 , -b„ a„ 1 ... 0,0 , , 0, ... a„_„ 1 , 0,0, Q-...-K, a„ if we expand (lll. 24) according to tlie elements of the last row and column. Similarly 2n = a. > 1 , 0, 0.. .0,0 -K a, , 1, 0.. .0,0 0., -63, a,, 1 .. .0,0 , 0,0, 0. ..«„_„ 1 , 0,0, 0. ..-&„, a„ Since p^ = b^-^, we can write the convergent in the form ^' ^, (^°^ ^'•)- 4. The determinants of the form q^ have been called con- tinuants by Mr Muir. Since if M„ is the number of terms in the continuant of order n an equation of differences which gives Since u^=\, % = 2, we have M„= {(1 + VS)"^' - (1 - V5)""'} - 2"^' V5. 172 THEORY OF DETERMINANTS. [chap. XIII. It is easy to shew by the binomial theorem that this number is an integer. Prof. Sylvester obtains this number in the form of the series 5. The value of the continuant q^ is the same as that of the determinant d^, a„ c, ... 0, d„ a^ ... provided only cA+i = 0, 0, ... a„ ■■-K,,.ir=l,2...n-1). This is clear' if we expand by iii. 24, according to the elements which stand in the last row and column. For then ?»' = «»2'„-i - d„G„_,q'„_^ while ql = q^, q^ = q^. Hence q^ = q„, the equation of differences being linear. Thus we can also write in 1, , K a^ . - 1> ... 0, ^3 . ^8 < -1 ... 0, 0,6,, a, ... 6. The value of the continued fraction is not altered if we replace by kb^, ka^, kh^^^. For the quotient ^ is unaltered if we multiply numerator and denominator by any the same number. If we multiply both by k, the row • •• -6,, a,., 1 ... 4 — 8.] THEORY OF CONTINUED FRACTIONS. 173 in each is replaced by ... — kh^, ka^, k ... and by Art. 5, in place of the last k, we can write unity if we replace b^^ by k\^^. Since then we can write the continued fraction in the form ,~0„ h, : ^ a.-i^n h->f k+ k+ '" + k q„ can be written in the form of the skew determinant k , a, , Q ,Q -Oj, k , a^ , ,-a„ k ,a, , ,-a„k where '■"Jm Thus the convergents to a continued fraction can always be represented by the quotient of two skew determinants. 7. In any determinant D we have d'^D dD dD dD dD D da^^da„„ da^^ da^, da,^ da^x For D take the continuant g„ (Art. 5) , then (f J _ 1 da„da„„ b, ' ^" dD_ da„„ ^"-" da,. 1 ^^=6A...&„, ^ = (-1)" da, da. Thus §'„p„_i - '7„_,i3„ = (- ly" KK ■■■K- 8. In the case of the ascending continued fraction Vt K± a. a.. 174 THEORY OF DETERMINANTS. [chap. XIII. it is clear that if the n* convergent be — , the scale of relation is ^ In Hence To determine p„ we have the system of equations : Pi =h - ^n-lPn-i +Pn.l = &„-l The determinant of this system is unity, all the elements to the right of the leading diagonal vanishing ; •• Pn 1,0, ... , b, -a„ 1, 0... , b, ,-a,,l... , 63 0, .. • 1 . K-. 0, 0.. •-««. b„ Multiply all the columns except the last by — 1, and move the last column to the first place ; the determinant is unchanged, thus p,= \, -1, ...0, b„ a,, -1 ...0, 63, , a, ... 0, b„, 0, ... 0, a„ The m'" convergent to the fraction is Pn The number of terms in p^^ is n. 8,9.] THEORY OF CONTINUED FRACTIONS. 175 9. By means of these determinant expressions for the conver- gents we can transform an ascending continued fraction into a descending continued fraction. In the determinant p„ of the preceding article multiply the r'" row, beginning with the last, by b^_^, and subtract from it the (r— 1)" row multiplied by 6,., and do this for all the rows. The determinant is altered by the factor k={b,b,...b„J-\ and p=k b„ -1 , 0, aj)^+b„ -b, 0, -aj),, aj},+b. 0, , 0, , 0, , ■■an- -A-&b„_^, - b -a„_A-i .«n-A- , -a„ Similarly, since qn = a^cl,. ■• «n = a„ -1, ... , 0, a,, -1... , 0, , % ... , 0, 0, 0, 0, ...a„_„ -1 ... , a„ qn = ^ a, , -I , -afi^, aj>, + b^, -b^ , -"A ' <^A + K - «n-A. aX-1 + b„ Now on inspection it is clear that these determinants p^ and 17C THEORY OF DETERMINANTS. [chap. XIII. q„ are continuants as defined in Art, 3, whose 2"", 3"* ... (n — \'f rows have been multiplied by 6, , 62 . . . 6„_2 respectively, also ^■'"i:- Whence by Arts. 3 and 6 aX aj}, + b. aAK a„-A-3&»-i C^n-A-A ,+63 '"' a»-A.-2 + ^«-i- aJ>n-i+K' which gives us a rule for transforming an ascending continued fraction into a descending continued fraction, the number of quotients in each being the same. 10. We can make immediate use of this theorem to deduce a formula of Euler's, by means of which a series can be converted into a continued fraction. Take the series S=A^-A^ + A,-A^ + A^, 1, 0, 0... A„ 1, 1, 0... A„ 0, 1, 1 ... + (-1)"-'^ A„, 0, 0, ... 1 as we see by subtracting from each row the one below it, beginning with the last, when the determinant reduces to its principal term. Multiplying each column after the first by — 1, we reduce the de- terminant to the continuant for an ascending continued fractioq. Thus the above series is equal to : -1 -1' (-1)" 1 -1 and transforming this by the rule just obtained to a descending continued fraction A A, A A ^ '^ 1-A,-A,+ A, A„-iA„ A. A„ A A ■A^+ ••• A^-A„ 1+ A^- A^+ A^-A,+ ■■■ A„_,-A„' 9 — ll.J THEORY OF CONTINUED FRACTIONS. 177 If the original series is we can obtain its form as a continued fraction by altering the con- tinuant to S in accordance with Art. 6, when we get 1 ^.' A,' 11. Various generalisations of continued fractions have been devised by Jacobi and others. The following generalisation, due to Fiirstenau, is taken from a review of his memoir by Giinther. If X and y are any two real numbers, and we write x, X. 2/ = «o + ^. 3'i = »i + ^. J. , 1 I. 1 a; = 6„ + — , a? = 6 + — 2/i y. y^ = a^ + — ■■ 1 x„ = h„ + y^ where a and 6 are the greatest integers contained in x and y, then on substituting we have : 2/ = a„ + + ■ + + + ■ a. + — + + - + ■ and x = h + ■ 1 + — ft. b, + — ''4 + + ft. + 1 + a. + + S. D. 12 178 THEORY OF DETERMINANTS. [chap. XIII. If now all that stands to the left of one of the vertical lines be called a first, second . . . convergent, and if we denote the numera- tors of X and y by X^, Y^, while the denominator, which is clearly the same for both, is called iV"^, we shall have (7, X, N),^, = a,,, ( Y, X, N), + 6,,. ( Y, X, N)^^^ + {Y, X. X),^. Thus the equations have four instead of three terms, and we get y.= «o . K. 1 . . -1, a.. K 1 . . 0, -1 «2 h- .. 0, 0, . ■ % ^,= K. 1 , 0, . . -1, «.. h. 1 . . , -1, «2. ^3- . , , . . a. K = «1. K> 1, . .. -1, a,, h. 1 . .. , -1, a„ K- .. , , 0, . •• a. Corresponding to the theorem of Art. 7 we have now Y Y Y Y Y Y p+1' r> p-i = 1. 12. If ordinary continued fractions be called fractions of the first class, those in Art. 11 may be called fractions of the second class. Furstenau extends the idea still further, and summing up his results we may state them as follows : If we seek to determine n quantities x^, x^ ... a;^ as fractions of the form -^1 ^^ x„ X' -.=^- ■ ''-~ N 11, 12.] THEOEY OF CONTINUED FEAOTIONS. 179 each such fraction can be written as a continued fraction of the (w—l)* class. Thep"" convergents to these continued fractions take the form and if Xp, X^^ X. N,' N, •■■ N,' .. a„ ... a„. are the quotients entering into the continued fractions, then -^M ~ '^ip-^P-ii + ^ap-^P-ss + + ^n Hp'^p-n-lq > The quotients X and JVare always connected by the equation Pi.' -^P-ll' -^i>-21 ••• ^^p-nl ^P2> ^P-li 'Xp- x^ ■p-2n ' T p-i ■ P' = (- 1)"^. The author also shews that the real roots of an equation of the n'" order can be represented as periodic continued fractions of the (m - !)"■ class. 12—2 CHAPTER Xiy. APPLICATIONS TO GEOMETRY. 1. The axes being rectangular let the co-ordinates of the angular points of a triangle ABC be (x^, y^ {x^, y^ (a;,, y^. Then if A is the area of the triangle it is plain from the figure that A = trap. BN — trap. BL — trap. CL * =4(^2+ 2/3) (^2 - «s) - i- (J2 + yd K - «i) - i ii/s + yJ K - ^>^' or 2A = y^x^ - y^x^ + x^y^ -x,y, + x^y^ - x^y^ 1, \ 1 = 1. a'l. 2/1 x^, *2. ^i ]. «2. y. Vv y-v y. 1' ^3. 2/3 If the axes were oblique this would have. to be multiplied by the sine of the angle between the axes. Thus 2A=sin(Xr) I, 1, 1 yp y.. ya 1,2.] APPLICATIONS TO GEOMETRY. 181 where (XY) is the angle between the axes. This form is however not often used, and unless the fact is specially mentioned the axes are supposed to be rectangular. If we multiply the first row by x^ and subtract it from the second, then the first row by y^ and subtract it from the third, we get 2A= ir.-Xj, x^-x^ 2/2 - Vi' ys - 2/1 It must be noticed that the area of a triangle changes sign if we alter the cyclical order of the letters. Thus AB G and A CB are equal triangles, whose areas are opposite in sign; ABG and BOA are equal in magnitude and agree in sign. 2. Let the co-ordinates of the angular points of a tetrahedron ABCD be {x^, y^, z^ ... {x^, y^, z^. Let Fbe its volume. Let A be the area of the triangle BCD, and let the equation of its plane be {x — x^ cos a + (2/ — 2/2) cos /3 + (a — z^ cos 7=0. The projection of the triangle BCD on the plane of xy is A cos 7, and the co-ordinates of its angular points are (^2. 2/2) («s. 2/3) (^4. yJ; thus, by Art. 1, 2A cos •y=x, Similarly we get 2/3-2/2' 2/4-^2 2A cos /3 = ^S~ ■^2' ^i X^ X , X •^2 ■oo„ 2A cos a : ^3 — ■^2' ^4 — ^2 1{ p is the perpendicular from A on the plane BCD, -P = (^1 - ^2) cos a -1- (y, -y.,) cos ^ + (z,- z^) cos 7. Hence -6F=-2Ap = 2A cos a (x^ - x^) + 2A cos /3 (y^ - yj -h 2 A cos y (z^ - 2,) = {x^-x,) ys-y^' y^-y. + (yi-y,) x^ — x^, x^ x^ + («, - ^2) ^3-^2. 2/4-2/2 182 THEORY OF DETKRMINANTS. [chap. XIV. "^1-^2' ^3-^2> *4-'^2 2/1-^2. 2/3-2/1!' 2/4-2/2 ^1 - ^2. ^3 ~ ■^2> ^4 - ^2 1, 1, 1, 1 «,-«2. 0, «a-«2' «'4-'»2 2/1-2/2. 0. ^3-2/2. 2/4-^2 ^2. 0> ^S , -3o Or if in this last determinant we multiply the first row by ^2' 2/2' ^^2 ^'^'^ ^<^^ i* *^ *^® second, third and fourth rows re- spectively, 6F= 1, 1, 1, 1 •^11 •"2' ■''s' ■"4 yi. 2/2> 2/3. 3^4 •^JI ■^2_> ^3> ^4 3. If the tetrahedron be referred to oblique axes through the same origin, and if the cosines of the angles these make with the rectangular axes be given by the scheme X F Z X k K h y m, m. 7»3 z ^1 ^ ^3 x = Xl^+Yl, + Zl^, &c. 1, 1, 1, 1 = 1, 1. 1, 1 1, 0, 0, ^1> ^2) '''s' ^4 -^1' ^2' -^8' ^4 0, K, ??i„ ^ 2/.. 2/2. 2/3- 2/4 y T F ^1' ^2' -'3' 5^4 0, K, m„ ^ ^l> ^2' ^3' ^4 ^1> -^2' -^3' ^4 0, K, W^a. «3 Now let D = h> mj, '^i K> m,, Tlj, «3. m,, ^ Then remembering that /,Zj + TO,mj + n,Wj= cos XF, &c.; 2— 4.J we have 183 APPLICATIONS TO GEOMETRY. D'= 1, cosXY, cosXZ COS YX, 1, COS YZ cosZX, cos ZY, 1 This determinant is usually called the square of the sine of the solid angle, contained by the oblique axes in analogy with the determinant sin'XF = in a plane. Thus 1, cosZF cos YX, 1 D' = siTi'{X YZ). And in oblique co-ordinates 6F= 1, 1, 1, 1 sin(Zr^). Xj, X^, Xg, X^ Y Y Y Y ^i> ^2> ^a> ^i 4. From the determinant expressions in Arts. 1 and 2 we can at once write down a number of geometrical relations. If the distances x be measured along a straight line from a fixed point, we see that 1, a;, 1. a'* K-^<) = ('^') is the distance between the two points marked k and i. The determinant 1, ^1. 1, ^, 1, «'2. 1, ^2 1, a'a. 1, ^9 1, x„ 1, ^'i vanishes identically, because it has several columns alike. Ex- panding it by III. 6 according to products of minors from the first two and lasli two columns, we get (12) (34) + (13) (42) + (14) (23) = 0. 184 THEORY OF DETERMINANTS. [chap. XIV. Or, if we call the points A, B,C, D, this is the well-known relation between the segments formed by four collinear points AB.GD + AG.DB + AD.BG=Q. If we expand the vanishing determinant I 1, a-i, Vv 1, !«0 Vi 1 (z=l, 2...6) according to minors from the first three and last three columns, we get no geometrical relation, the terms cancelling each other in pairs. But if we expand the determinant \l,x„y„z„l,x^,y„z,\^0 (i = l, 2...8) according to the products of minors from the first and last four columns we get an identical relation of thirty-five terms between the volumes of the_ tetrahedra, formed by eight points. 5. Again, for five points 1, 1, 1, 1, 1 =0. 1, 1, 1, 1, 1 f)ft /yi ryt n/t /v» '*'l) ■'2> -^a' 4' "^5 yi. 2/2. 2/3. Vo y. If ■Wj = volume of tetrahedron (2345) and we expand the deter- minant according to the elements of the first row, by iii. 10, we get V^+V^ + V^ + V^-lrV^= 0. 6. By the theor.em v. 4, 1, 1, 1 *i! *2' Va VV ^2' ^3 1, 1, 1 I, 1, 1 ?1> ^8' ^3 Vn 2/2. Vs Or if the two sets of three points be called ABC, BEF, ABCx BEF ^ABEx FBC +AEFx BBC + AFB x BCE is a relation between triangles. 1, 1, 1 6l> 62' S3 Vv V2, V3 1, 1, 1 •"'IJ 92> &3 1, 1, 1 S3> ^2' ^3 'Ja. yn. 2/3 1, 1, 1 '"l' 63' tl 2/1. '73. % 1, 1, 1 %> ^2. ys 4—7.] APPLICATIONS TO GEOMETRr. 185 The product of the two determinants 1, 1, 1, 1 Vi^ '?„ '73. V, can be represented either as a sum of four terms 1, 1, 1, 1 ^1) ^2' ^a' ^4 2/i. 2/,. 2^3' 2/4 ^l> ■^2. •2'3> ^^4 1, 1, 1, 1 ^1> *2' ^3> Si 2/i' 2/2. 2/3. ■^i ^1> ^2' ^3' fei or as the sum of six terms 1, 1, 1, 1 ^1' *2' Si' ?2 2/.. 2/2. •^i' '72 ■^1> ^2' ?1J ?i 1, 1, 1, 1 f2. I'3> f4> ^4 '72' '7 3. '74' 2/4 b2' ts' fe4' ^4 1, 1, 1, 1 S3) S4' ^3' ^4 '73' '74' 2/3' 2/4 bs ' b4 ' '^3 ' ^4 + + Or calling the two sets of points ABGD, EFGH, we have the identical relations between the volumes of tetrahedra : ABGD X EFGH= ABGE x FOHD - ABGF x GHED + ABGG X HEFD - ABGE x FGED ABGD X EFGH = ABEF x GHGD + ABGH x EFGD +■ ABEG X EFGD + ABEFx EGGD + ABEE XFGGD + ABFG x EECD. Application of Alternate Numbers in Geometry. 7. In applying alternate numbers to geometry, a number stands for a point in a flat space whose dimensions are one less than the number of units. To begin with a plane, the units e^, e^, e^ stand for the vertices of a fundamental triangle ABC. Anj other number F= xe^ + 2/e, + 2:63 186 THEOEY OF DETERMINANTS. [CHAP. XIV. stands for some point in the plane of the triangle. It is generally convenient to assume that so that X, y, z may be taken to mean the ratios of the triangles PBG, PGA, PAB to the triangle ABC, though this is not neces- sary. If P and Q are two points, then mP +nQ m + n is a point in the line PQ, dividing PQ in the ratio m : n. Thus \{P+Q) is the middle point, and P—Q the point at infinity of PQ. Similar definitions hold for a space of three dimensions. Four points ABCD being taken and represented by the units gj, fig, gg, e^ any other point in the space is represented by P = xe, + ye^ + ze^ + we,, where if we choose we may write x-\-y-\-z-\-w=\, X being the ratio of the tetrahedron PBGD to ABCD. And so on for a space of any number of dimensions. Then a binary product e^e^ is a unit length measured on the line joining the points e^, e^ or the distance between the points A ternary product e^e^e, is a unit area measured on the plane of the points e^, e„ e„ or the area of the triangle formed by the points e,., e^, e,. And so on. In a space of two dimensions the product of three points is the area of the triangle they form referred to the fundamental triangle. Now if P= x^e^ + y,e^ + z^e^, Q=«A+ ••• R = x^e^+ ... PQB = ^'l. Vv ^. «2> y.. ^» x„ y.' «3 e,e„e. 7,8.] APPLICATIONS TO GEOMETRY. Ib7 And 6,6^63 = ABC = A, the area of the fundamental triangle, so that in areal co-ordinates PQR = ^2' 2/2' ^2 Similarly in a flat space of three dimensions if 6,6,636,= V is the volume of the fundamental tetrahedron, the volume of the tetrahedron formed by four points is PQRS: ^'i. y,. ^.. M>, *2. y^' ^2> W-s ^s' 2/3. ^3> ^s «4. ^4. «4. W4 y. Similar definitions may he stated with reference to flat spaces of more than three dimensions. The assumption which has been made throughout the present work, that the product of all the units of a system is unity, receives here its justification and explanation. For, geometrically speaking, the product of the units is the measure of the funda- mental figure of the space considered, which is our unit of measure. In a plane, for example, it is the area of the triangle of reference, in ordinary space of three dimensions the volume of the tetrahedron of reference. It is no part of the plan of the present treatise to develop the geometrical applications of alter- nate numbers ; for these we must refer to the memoirs and works of Grassmann and Schlegel. Angles between straight lines. Solid angles. Spherical Jigures, 8. With rectangular axes let ?,, m,, «., X,, M,, v, ?,, w,. '^^ \' /*2' "a 188 THEORY OF DETERMINANTS. [CHAP. XIV. be the direction cosines of two sets of straight lines, then if cos {ik) = IXj^ + m.iJL^ + n.v^ is the cosine of the angle between the r"" line of the first and ¥^ of the second system ; if we compound the two arrays, we get the determinant I cos {ih) I . Hence by iv. 2, if there are two sets of four straight lines we get cos (11) ...cos (14) =0 (i). cos (41) ... cos (44) If there are two sets of three straight lines a, &, c ; /, g, h, cos af, cos ag, cos ah cos hf, cos hg, cos hh cos cf, cos eg, cos ch h< m„n^ h. m^, «2 h, «*s. '^S = h'^1 \. /^l h> '^2 \,f^2 = sin (ahc) sin (fgh) (ii) If there are only two straight lines in each set I cos (11), cos (12) I cos (21), cos (22) Now if n, V be the directions of the shortest distances between the lines of each pair, 0, , the angles between the pairs cos (11), cos (12) cos (21), cos (22) = sin 6 cos {nz), &o. =■ sin 6 sin ^ cos (nv) (iii) 9. If in the relation (i) of Art. 8 the two sets of straight lines coincide with one set of straight lines a, b, c, d, we have 1 , cos {ah), cos {ac), cos {ad) cos (&a), 1 , cos (6c), cos (6cZ) cos {ca), cos (c6), 1 , cos {cd) cos {da), cos {dh), cos {dc), 1 = 0. 8— 11. J APPLICATIONS TO GEOMETRY. 189 This is the identical relation between the mutual inclination of four straight lines in space, or also the relation between the sides and diagonals of a spherical quadrilateral. If we write — cos {AB) for cos {ah), or what comes to the same thing change the signs of the elements in the leading diagonal, it becomes the identical relation between the cosines of the dihedral angles of a tetrahedron formed by four planes A, B, C, D perpen- dicular to the lines a, b, c, d. 10. If the two straight lines marked 1 coincide with two straight lines u, v ; while those marked 2, 3, 4 coincide with a set of oblique axes x, y, z, cos uv, cos ux, cos uy, cos uz = 0, cosa;i;, 1 , cosxy, cosxz cos^v, cosyiB, 1 , cos yz cos zv, cos zx, cos zy, 1 which gives the cosine of the angle between two straight lines u, v, referred to a set of oblique axes x, y, z in terms of their direction cosines. 11. As another example of the use of the same formula, let ABO, A'B'C be two spherical triangles, 0, 0' the centres of the small circles circumscribing them. For our two sets of straight lines take the lines joining the centre to O'ABG, A'B'C. Then if 00' = ^, and R, R' are the radii of the circumscribing circles, we get cos ^, cos R', cos R', cos R' = 0. cosiJ, cos{AA'), COS [AB'), cos{AG') cobR, cos {BA'), cos (BB), cos (BC) cosR, cos {CA'), cos {GB'), cos (CC) We can write this cos(lism{AB(J)sin{A'B'C')=-cosRcosR' 0, 1 ... 1 1, cos^J.'...cos(^C") l,cos(C^')---cos((7(7') . 190 THEORY OF DETERMINANTS. [chap. XIV. If the angle at which the small circles cut is ■>fr cos f^p< ^f 11 — 13.] APPLICATIONS TO GEOMETRY. Hence, by IV. 2, for two sets of five straight lines sin' 1(11) ...sin' ^(15) =0 sin'^(51)...sin'^(55) For two sets of four straight lines a, b, c, d; a, b', c', d', 191 (i). 16 sin'^(aa') . . . sin"^ {ad') sm^(da')...sm'i{dd') = - 1 1, ?,, VI., wj X 1 1, \, fi., v^ I (i=l, 2;3, 4) (ii). Expanding the determinants on the right according to the elements of their first column, our determinant = {sin Q)cd) + sin {cad) + sin {abd) — sin {abc)] X {sin {b'c'd') + sin (c'a'd') + sin {a'b'd') — sin (a'b'c')}. For two sets of three straight lines, our determinant is 1- cos (11) ... 1- cos (13) 1- cos (31) ... 1- cos (33) or 1, ... 1, 1- cos (11)... 1- cos (13) 1, 1, 1- cos (31) ... 1- cos (33) 1, -1, 1, -cos (11) 1, 1, - COS (31) -1 -COS (13) • COS (33) This is equal to the sum of the products of determinants of the third order taken from the two arrays. Omitting the term -cos (11) ... -cos (13) h' *^1' \ h' "^2. ^S l„ m,, W3 -\> -^1> -"1 -\> -^2' -^2 -cos (31) ... -cos (3.S) we get 0, 1 ... 1 1, cos (11).. .cos (13) = |l,Z,m||l,\,^| + U,^,?i||l,\,v| + \l,m,n\\l,/j,,vl 1, cos (31)... cos (33) If the straight lines be called a,b,c; a, b\ c, and N^, X^, R^ 192 THEORY OF DETERMINANTS. [chap. XIV. are the directions of the shortest distaaces between he, ca, ah, we have 1 1, Z, m I = sin (he) cos {N^z) + sin {ca) cos {N^z) + sin {ah) cos {N^), 1 1, X, /tt I = sin(6'c')cos(iV»+sin(c'a')cos(iV»+sin(a'&')c,os(A^»,, and similarly for the other determinants. In particular, if ahc lie in one plane, and a!h'c' in another, the normals to the two planes being N, N', the value of the determinant is {sin (6c) +sin {ca) + sin {ah)] {sin (6'c') +sin {c'a) + sin {a'h')] cos {NN'), viz. this 0, 1 ... 1 1, cos {ad) ... cos {ac') 1, cos {ca') ... cos {cc') (iii). For two sets of two straight lines we deduce in the same way, if B, r are the directions of the external bisectors between them. 0, 1, 1 1, cos (11), cos (12) 1, cos (21), cos (22) . . ah . a'h' ,„ , • 4 sin -^ sm -^ . cos {Mr). 14. If we compound the arrays l^, m^, M„ 1, \, fi^, i/j, 0, 1 0, 0, 0, 0, 1 \> h-o Vi, 0, 1 0, 0, 0, 1, 0, we get the determinar t cos (11) . . cos (ii), 1 cos {il) . 1 . cos {ii), 1 1, • Hence for two sets of five straight lines cos (11) ... cos (15), 1 cos (51) ... cos (55), 1 1 ... 1 = 0. 13—16.] APPLICATIONS TO GEOMETRY. 193 For two sets of four lines cos (11) ...cos (14), 1 = -\l,l,m,n\\l,fi,\,v\, cos (41) ... cos (44), 1 1 ... 1, and so on. But these are not new theorems. In the first for example, if we expand by III. 24, according to products of elements in the last row and column, each term vanishes by Art. 8. 15. If On Systems of Straight lines. cos a cos /S cos 7 be the equations of a straight line, then a = cos a, b = cos /8, /= q r cosyS, cos 7 9 = r p cos 7, cos a A = c = cos 7, p q cos a, cos/8 are called the co-ordinates of the line. It is plain that af+ bff+ ch = 0. 16. If the constants belonging to two straight lines be denoted by the suffixes 1 and 2, the equation of a plane through the second line, parallel to the first, is COSOj, COSySj, C0S7j COSttj, COSySj, 005 7^ If d be the shortest distance between the two straight lines, and 6 the angle between them, it follows that dsm0= p,-p^, q,-q„ r^-r^ costtj, cos/3,, cos7j cosoj, cosyS,, cos7j Pi' ?i. ^1 cos a,, cos ,/S,, cos 7, COSO„, COSyS , COS 7 P2' ?2. ^ cosa,, cos/8j, cos7j cos a,, cosjSj, cos 7, = aJi + Kffi + cA + «i/2 +^15^^ + Ci^, S.D. 13 194 THEORY OF DETERMINANTS. [chap. XIV. If the expression on the right vanishes, then either d = 0, i.e. the two straight lines intersect, or sin^ = when they are parallel, and hence also meet. It is convenient to have a name for the expression on the right. If a unit force acted in one of the lines its moment about the other would be dsin^, i.e. in terms of the co-ordinates of the lines a J, + h9, + ^K + ^2/1 + ^2.9i + cA- Hence we shall call this the moment of the two straight lines. If two straight lines meet their moment vanishes. 17. Let us take two systems of straight lines whose co- ordinates are «!. &i, Ci,/i, ^Ti, /?! //, gi, K, al, hi, Ci' «i. ^. C(,/i, S'i. K /•'. S'.'. K^ <- K' K- Then if m^, denotes the moment of the line r of the first and s of the second system, by compounding the two arrays we get the determinant i "»* I Hence for two sets of seven straight lines !„ ... m„ =0, m„ ... m„ an identical relation between the mutual moments of two sets of seven straight lines. If the two systems coincide 0, m^. fti^. m„, m,, ... For two sets of six straight lines 0. m,. .. m„ w., m„ = |a., 6., c,,f.,g.,h.\ >^\f!>9!>h-,a:,b:,c:\ (^ = l,2...6). If one of the sets of six straight lines — say the first — is met by a common transversal ■whose co-ordinates are a, h, c, f, g, h, we have for each of the straight lines of that system of, + ^9i + '^K +/«i + 9^i + h(^i = 0- 16—18.] APPLICATIONS TO GEOMETRY. 195 Thus the first of the determinants on the right vanishes, and mil ••• ^16 = is the relation between the mutual moments of the two sets of six straight lines, one set of which is met by a common trans- versal. If the two sets coincide we get the identity for a system of six lines met by a common transversal. 18. If the moments of a system of forces about one set of seven lines be m^, m^ ... m,, and about a second set j!j, w^ ... %,, we can establish an identity among the moments involved. For if any force P of the system act in a line whose co-ordinates are a, h, c,f, g, h, we have THi = SP {a/l + bg-^ + c\ +fa^ + gh^ + hc^ =f,lPa+g,-ZPb + ?i,%Pc + a,tPf+ \tPg + c,^Pli, and six other equations for m^ ... to,. Hence eliminating SPa, tPb...tPh, we get wii, ttj, 6i,Ci,/i, 5^1, ^ =0, and a similar equation for the other system. Hence each of the determinants 0, TO,, «!, 61, Ci, /i, gi, K 0, TO,, a„ &„ c„f,,g,, \ 1, 0, 0, 0, 0, 0, 0, ni, ^,fi,gi, h'> «i'. W> c/ «,> 0,//, 5^/, V. <- ^'c/ 0, 1, 0, 0, 0, 0, 0, vanishes. Forming their product we get TO„ ... m,„ w, TO„ ... TO„, n, m, ... TO,, = 0. 13—2 196 THEORY OF DETERMINANTS. [chap. XIV. Tetrahedra and Triangles. 19. Let there be two systems of points in space whose co- ordinates referred to rectangular axes are {x^, y^, aj, (f ., t)., f.). Let us compound the two arrays x„y„z„\,0 -2^^,-2,7., -2^0, 1 0, 0, 0, 0, 1 0, 0, 0, 1, 0, we obtain the determinant "u- •Ci,. 1 Ca • ■ c„, 1 1 . . 1 where c„ = - 2«^f . - ly^), -2zX,. To the r"" row add the last multiplied by x^ + y^ + z^, and to the s"" column add the last multiplied by ^^ + r,^ + f/, the deter- minant is unaltered and its elements are now i.e. d^^ is the square of the distance between the r'" point of the first and s* point of the second system. We have then the deter- minant d„ ... d„, 1 4 ... d,„ 1 1 ... 1 If i = 5 the determinant vanishes, hence d,,...d,„ 1 =0. d. ^55-1 ■(i) is the identical relation which subsists between the lines joining two sets of five points in space. If the two systems coincide d., = 0, and the determinant, which is then symmetrical, gives the relation between the lines joining five points in space. The relation in this form is due to Cayley. 19.] APPLICATIONS TO GEOMETRY. 197 ^u ••• d,„ 1 "i. Vv ^v 1. -2|,, -2,7,, -2?,, 0, 1 -2?., -27., -2r,. 0, 1 , , , 1, d^^ ... d^, 1 x^, y^, z^, 1, 1 ... 1 , 0, 0, 0, 0, 1 = 2887F' (ii), where V, V are tlie volumes of the tetrahedra formed by the two sets of four points. If the two sets coincide in a single tetrahedron, for which a, a' ; b,b'; c, c' are pairs of opposite edges, 2887'' = , c'^ b", a", 1 c\ 0, c% 6^ 1 V\ c\ 0, a\ 1 d\ b\ a\ 0,1 1, 1, 1, 1, If i = 3, we have d,....d,„ 1 <^8i — <^83> 1 =-^\x,yX[ \U,\\-^x,z,\\ l?,r,l|-4|2/,3,l| \%t\\. 1 ... 1, all the other determinants on the right vanish identically. Now if A, A' be the areas of the triangles formed by the two sets of three points, ?, m, w ; X, /i, v the direction cosines of the normals to their planes \x,y,\\ = '± projection of A on plane xy = 2Aw, and similarly for the others ; hence if ^ is the angle between the planes of the triangles J„ ... (?,3, 1 ^31 ••• '^SS. 1 1 ... 1, = -16AA'cos^ (iii). Lastly, if i = 2, «?„, d,2, 1 1,1,0 x^, 1, ^„ 1, 0, 0, 1 - 2?„ 0, 1 -2?,. 0, 1 , 1, + ... = 2 (a', - !c^ (I. - f») + 2 (y. - 2/,) (,;. - t,,) + 2 (^. - z^ (?, - ?J, 198 THEORY OF DETERMINANTS. [chap. XIV. the other terms vanish. Now if a, b be the lengths of the lines joining the points of the first and second systems and the angle between them, ^i~^a ?i ~ ?2 '-' + . + .=coad. Hence <^n' ^.2' 1 1, 1, = 2a&cos (iv). 20. If in case (iii) of Art. 19 we allow the two sets of three points to coincide with the vertices of a single triangle whose sides are a, b, c, -16A^= 0, c\ b\ 1 c', , a^ 1 b\ a^ 0, 1 1, 1, 1, Multiply each column by abc, then -16A''a*6V= , abc\ ab\ abc abc', , a^bc, abc a¥c, a'bc, , abc abc , abc , abc , Divide the first, second, and third rows and columns by be, ca, ab respectively, then - IGA'' = 0, c, b, a c, 0, a, b b, a, 0, c a, b, c, a, b, c, b, a, 0, c c, 0, a, b 0, c, b, a by an interchange of columns. If in the first expression for — 16A' we divide the second and 19—22.] APPLICATIONS TO GEOMETRY. 199 third columns by a", and then multiply the first and last rows by a', we get : 16A'= 0, c\ h\ a' c\ 0, 1, 1 b\ 1, 0, 1 a\ 1, 1, 21. If in case (ii) of Art. 19 one of the sets of four points- say the first — lies in a plane, V= 0, and d,„ 1 = 0. If one of the sets in case (iii) lies in a straight line the cor- responding triangle vanishes ; hence d„ ... d,, 1=0. By allowing the second system to coincide with the first we get the identical relations between the lines joining four coplanar and three collinear points. 22. In the identical relation c?„ ... d,„ 1 = 1 between the squares of the lines joining two sets of five points, let the fifth point of the first system be the centre of the sphere circumscribing the tetrahedron formed by the first four points of the second system, and the point 5 of the second system the centre of the sphere circumscribing the first four points of the first system. Then E'\ 200 THEORY OF DETERMINANTS. [chap. XIV. Also, if ^ be the angle at which the two circumscribing spheres intersect, d,, = E^ + R'^ + 2RB: coscj}. Hence with an interchange of rows and columns d.. dw 1' E' '^44> 1) ^^ 4 1 ... 1 , 0, 1 iJ^ ... iJ'S 1, d^ Multiply the fifth column by R" and subtract it from the last, and the fifth row by jR'" and subtract it from the last, then = 0. d,. d,. d,., 1, d^, 1, = 0. 1 ... 1, 0, 1 ... 0, 1, 2RR'coscl> Or, resolving according to the elements of the last row and column, we have by Art. 19 (ii) 57QVRr'R'cos^= cZ„ ... d, d,, ... d^ We see from this that so long as the circumscribing spheres remain fixed the tetrahedra can turn about in them without altering the value of the determinant on the right. The determi- nant vanishes if the circumscribing spheres of the two systems cut orthogonally. This relation is due to Siebeck. 23. If in Art. 22 we allow the two tetrahedra to coincide we get, since <^ = tt, lQ{6rRy = - 0, a'\ h'\ c' a'^ 0, c^ ¥ b", c^ 0, a^ c'^ b^ a^ Multiply the second, third and fourth rows and columns by a', V, ^ respectively, then 16(6Fii;fa*6V = - 0, (aa:)\ {hhj, (ccT {aa'f, 0, a^6V, a%'c^ [hhJ, a%V. 0, a'bV {cc'Y, a'bV, aVc\ 22—24] APPLICATIONS TO GEOMETRY. 201 Divide the second, third and fourth rows by {obey, then multiply the first column by the same quantity, 16(6FE)=' = - 0, {aa'y, (bb'r, (cc'y {aa'y, 0, 1, 1 {bb'y, 1, 0, 1 icc'y, 1, 1, Now if we write aa = kx, bh' = ky, cc' = kz, then if A is the area of the triangle, whose sides are x, y, z, we have by Art. 20, { 1 a,i- •^66' 1 1 . . 1 is an identical relation between any two sets of five points in space. If the ellipsoid becomes a sphere we regain Cayley's relation (Art. 19, i). For i = 4, we have 288 FF' <^n- ■t^M. 1 «41- ■ o^, 1 1 . . 1 26—28.] APPLICATIONS TO GEOMETRY. 205 V, V being the volumes of the tetrahedra formed by each set of four points. 27. The polar plane of a point P{x^, y^, z^ with respect to the ellipsoid, is ^. -t- J. + c" The distance of a point Q(^,, t?., Q from this plane is 1 cc^ y^ z' / a* ^ 6^ ^ c* ■ If (Q, P) and g- denote like quantities for the point Q, (P,Q)JQ,P)^^ x^l yr,^ zX p q a^ U' c" ' This function has been called by Faure the index of the two points P and Q, denote it by I^^. Then, by compounding the arrays whose e"" rows are we obtain 5 h 5 1. zl« ZJh Zli a' 6 ■ c' ' a ' b ' c ' = ^u- ■I. 4- •4 In- -lu 4- •4 36FF' 28. It may be remarked that these space relations connected with an ellipsoid are not reaUy more general than those connected with a sphere. For they are what the relations in an ordinary space become when the sphere x' + y'+z^ = R* becomes changed by a homogeneous pure strain to the ellipsoid x' y» z^ , 206 THEORY OF DETERMINANTS. [CHAP. XIV. Formulce relating to Systems of Spheres. 29. If r, s be the radii of two spheres, the angle at which they intersect, and d the distance between their centres, then d' = r' +s' + 2rs cos ^. The function 2rs cos is d!', the square of the distance between the points. If one of the spheres becomes a plane, and p is its distance from the centre of the other, J. P cos

is p. 30. Let (Xj, y., z) and {^^, 77^, Q be the co-ordinates of the centres of two spheres of radii r^ and p^, then if p.^ is their mutual power = x: + y: + z: - r: - 2x,^, - 2yj,, - 2z,^, + 1/ + v^ + c - p:- Hence, compounding the two arrays «i, 2/1, ^i> 1, «,' + 2/1' + ^i' - '"," x^,y^,z„ 1, x^ + y,^ + z^ - r^', and -2^,-2^.,-2r.,r + ^,^ + C-p,M -2^„-2v„-2^„^:+v:+c-p:,'i-. we see by iv. 2 that for two systems of six spheres Pu---Pu =0 Ps^---P^ .(i). 29—31.] APPLICATIONS TO GEOMETRY. 207 If COS (^^ is the cosine of the angle at which two spheres cut, we can also write this |cos<^^|=0 (i,k= 1,2 ...6). Por two systems, each of five spheres, Pn--Pu P^ .(ii) If the five spheres of one of the systems — say the first — have a common radical centre, taking this for origin we should have x' + y'' + z''-r' = c\ where c is the same for all the five spheres. Hence, in the first determinant on the right of (ii), the fourth and fifth columns are proportionals and the determinant vanishes. Thus Pn---Pu = .(iii) P,i--Ps: when the five spheres of one system have a common radical centre. If the five spheres of the first system reduce to points (iii) is the condition that they should lie on a sphere. If both systems reduce to points we regain Cayley's condition, that the five points of one system should lie on the same sphere. 31. But if neither of the determinants on the right of (ii) vanish, expand the first determinant with regard to the elements of the last column. Then i'i = a-,' + «// + ^i' - ''/ is the power of the origin (i.e. any point) with regard to the i"" sphere of the first system. Then if we write 1, 2, 3, 4, 5 for the centres of the five spheres, and denote by v,= (2345), v,= {S4>51), &c., the volumes of the tetrahedra formed by the points in brackets, and if accents denote similar quantities for the second determinant, we have in place of (ii) \p„\ = 288 (v,p, +v,p,-h... + v,p,) (v,>/ + . . . + v^p/) ii,k = 1,2 ... 5). 208 THEORY OF DETERMINANTS. [chap. XIV. Now describe about the origin a sphere of radius r, cutting the spheres r^ ... r^ at angles ^, ... .%2v^p. cos 0/ (i, k = 1 ... 5). Thus r'Z2v/r^ cos (fl^ is independent of the particular sphere r, let this be the orthotomic sphere of the first four, then this sum reduces to 2v^r^Rcos{rfi), and the second factor, in like manner, becomes 2v^' p^R' cos {p,R'). Hence Pn -P,5 = 1152t'3. Also if the radical axis of the spheres of the first system meet the plane of centres of the second system in P, whose power with reference to the spheres is p, and P', p' denote like quantities for the other system, 2RR' cos {EB') = PP" -p-p'. Hence p,,...p,, =16^A' cos (PP''-p-p'). 3i. If in the relations d,, ... d^„ I = - 288 FF', 1 ... 1 of Art. 19, we suppose the sets of points to be the centres of our spheres. Then if we multiply the last column by p^ and subtract it from the ■^ column, and the last row by r/ and subtract it from the k"" row, we get the relations = 0, Pu • -Pis' 1 Psl- ■• i'a' 1 1 . .. 1 S. D. 14 210 THEOEY OF DETERMINANTS. [chap. XIV. Pn ■ •■^.4. 1 Pu ■ ■■Pu' 1 1 . .. 1 = - 288 FF', which give relations between the mutual powers of two sets of five and four spheres. 85. Another element connected with two spheres is the length of their common tangent. For two spheres of radii r, s the dis- tance between whose centres is d and which cut at an angle ^, the square of the length of the common tangent is given by t^d'-ir-sY = 2rs cos'^ ^(f>. If one sphere reduce to a point, t is the power of that point with respect to the other sphere. If both spheres reduce to points, t is the square of the distance between them. 36. Using the same notation as in Art. 30, if t^^ is the square of the tangent common to the two spheres t. = («^< - ^.y + (y. - r,,r + {z, - Q' - (r - p,y = x,'+ y: + z: - rf- 2x,^,- 2yrn- 2zX,+ 2r^,+ ?,' + „,'+ C" ft'- Hence, compounding the two arrays ^1. Vv «i. '\. 1. a^i" + y^ + z^ - r^ «'i. 2/i. z„ r., 1, x,^ + y^ + z^ - r,' 0, 0, 0, 0, 0, 1 -21, - 2rj^, - 2?„ 2p^, ^,' + r,,^ + r,^ - p^\ 1 - 2f<, - 2^, ^ 2r<, 2ft, ^^ + r,^ + ?,» - ft», 1 0, 0, 0, 0, 1, 0, we get for two systems of six spheres the identity 1 ... 1 = 0. 34—37.] APPLICATIONS TO GEOMETBT. 211 For two systems of five spheres we should get = 576 iv,r, + ...+ V5) «ft + . • • + v^Ps), tn- •«.a. 1 t.- ■•<55> 1 1 . .. 1 using the notation of Art. 31. If t^ is the angle at which the plane of similitude of the first four spheres of the first system cuts each of these spheres, and (r^t^ the angle at which it cuts the fifth sphere, and similarly for the second system, we can reduce this to the form cos (p,T,)N tn- •«.. 1 ts.- ■%.> 1 1 . . 1 576v,r^v:p,(l- cosi. COST. Hence the determinant vanishes if one of the systems of five spheres has a common plane of similitude. For two sets of four spheres, after some reduction we can prove that in- K.> 1 ^41 • • • *«' 1 1 ... 1 = 288..' (1--^^), \ cos t COS T/ where ^ is the angle between the planes of similitude of the two systems, and t, t the angles at which they cut their sets of spheres. 37. By compounding the arrays whose i'" rows are and - 2^„ - 2v„ - 2^„ 2p„ |/ + 7,/ + ?f - p^ 1, we get the homogeneous relation between the sets of tangents common to two sets of seven spheres in- i.. <„ :0. 14-2 212 .THEORY OF DETEEMINANTS. [chap. XIV. 38. We may make use of this last relation to solve the problem: Determine the equation of the sphere having with five given spheres tangents of the same length. Let the equations of the five given spheres be s, = o ,Sf,=0. Take these for the first five of each set of spheres in Art. 38, let the sixth sphere be the one required, and the seventh a point on the sixth. Then we shall have t,, = 0, t„ = S, t,=k, and the equation is 0, t,,. ^13 > ^U> *15' *23> *24> ^26 > 1, 1, 8, *S1' *62' «=3. *M> 0, 1, s. 1, 1, 1, 1, 1, 0, ^1> ^2' ^3> ^i> S^> 0) This is apparently of the fourth order, but by means of the sixth rows and columns we can get rid of the terms of the second degree in the seventh row and column. 39. All the equations of this section relating to spheres are capable of numerous and varied applications, some of these will be found in the examples, and others in the memoirs of Bauer, Darboux and Frobenius. EXAMPLES. Prove the following relations : 1 — 5. 1. {b+oy, ab , ac , (b + cy, 6" , ab , {c + ay, be , {c+ay, a" , ao be {a + by b' a' {a+by 1 , 1 , 1 tan A , tan B, tan C sin 2 A, ii A, B, G are the angles of a triangle. 3. 1,. X, sin 2B, sin 20 {a + x) ^{c + x) {a + y) J{e+y) = 2abc {a + b + c)', = 2{bo + ca + aby. = 0, = 0, 1, z, {a + z) J{c + a) 4. 1 , cosa , cos(a+y8), C0s(a+/3+y), cos(a+y8+y+8) cos a , 1 , cos/3 , cos(/3 + y), cos(^+y+8) cos (a + /3) , cos )8 , 1 , cos y , cos (y + S) cos (a + 18 + y) , cos (;8 + y) , cos 7 , 1 , cos 8 cos(a+j8+y+8), cos(/S+y+8), cos(y+8), cosS , 1 5. a + b + c + d, a — b-c + d, a-b + c-d a — b-G + d, a + b + c + d, a + b — c — d a — b + c — d, a + b— c — d, a+b + c + d = 16 {bcd+ acd + o,bd + abc). = 0. 214 THEORY OF DETERMINANTS. [ex. 6. If a, I, c are the sides of a triangle of area A, 28 = a + h + o, then (b + c)", ah , ac , a ah , {e+af, he , h ac , he , (a+by, c a , h , c = -l6sA{a\+h\+c\), T-j, r^, r^ being the radii of the escribed circles. If the elements in the principal diagonal are (6 - cf, &c., the other elements being as before, the value of the determinant is (6+«n ah , ac , a ah , {c + ay, be , h ac , be , (a + by, c 1 , 1 , 1 {b+cy, ah , ac , 1 ah , (e + ay, he , 1 ac , be , {a + hy, 1 1 , 1 , 1 = - 1 6s A (flw, + hr^ + cr^), = 16A='-20a&cs. 7. If S=a^+a^+ ... + a„, Af = S~ a^, prove the following theorems : ■ x{x-sy-\ = {x + {n-2)S]{x-Sy-\ 1 ' 8. The determinant a, b, h, h . a, b, a, a . b, b, a, b . a, a, a, b (the diagonal consisting of a and 6 alternately and each row being filled up with the other letter) is equal to {-iy-^n-l){a-by'. The determinant is supposed to have 2w rows. 6 — 13.] EXAMPLES ON THE METHODS OF THE TEXT. 215 9. If in a determinant all the minors of the second order are divisible by the same quantity p, then the minors of the m"" order are divisible by^""'. 10. If in a determinant of the w* order there be a block of p by ^ elements all of which are divisible by a, the determinant is divisible by a"*^-". 11. Prove the theorems : a, b, c, «, a + b, a+ b + c, a, 2a + b, 3a + 26 + c, a, 3a + b, 6a + 36 + c, d, a+ b + c +d, 4a + 36 + 2c + d, 10a + 66 + 3c + (f, = a" O', b, c, d ... a, a+b, a+26+c, a+ 3b + 3c+d ... a, 2a+b, 4a+46+c, 8a+ 126 + 6c+d! ... a, 3a+6, 9a+66+c, 27a+27b+9c+d ... = a''r (n-l), where a, b, c, d ... are any quantities whatever, and -n is the order of the determinant. In the first determinant each row after the first is obtained from the preceding by the rule that the r'" element of any row is the sum of the first r elements of the preceding row. In the second determinant the r* element of any row is the sum of the first r elements of the preceding row multiplied respectively by the coefficients in the expansion of (l + xy~\ 12. If D = a, h c, d ... -a, b, P, q ... -a, -b, c, r ... -a, -b, -e, d ... (ji rows), then D^ 2"-' abed... The elements of the first row and leading diagonal are a, b, c, d ...; in each column the elements below the leading diagonal are equal to the element in the first row but of opposite sign, the others are any what- ever. 13. It I> = cosMttji, cos (m-1) a„ ... COSa„, 1 coswoj, cos(w-l)aj ...cosoj, I COSWa^, COs(?» — l)a„ ... COSa„, 1 216 THEOET OF DETERMINANTS. [ex. D. i>,= COs"a„, cos"'''a(| ... COSa^, 1 COs'aj, COs"~'a, ... COSOj, 1 COS a , COS " 'a ... COS a , 1 then sin(ji + l)a„, sin »ia„ . . . sin a„ sin(n+l)aj, sin wOj . . . sin a^ sin (w + l)a_ , sin na^ , . . sin a_ J\ «{n-l ) T\ n(ri+l) --^ = 2 ^ , yf-^ ' sin a„ sin a, . . . sin a_,. 14. If 6„ = (au + aa+ ... +aJ-o„, then 6u ... 6,„ *„i ■■■Kn :(-ir'(n-l) "11 ••• ""In But if fr., = («,, + «„ +...+»,„) -2a,. ill - K 6,. ... 6.. = n(-2)"- 15. Prove that every power of a symmetrical determinant is again a symmetrical determinant. 16. If for each element a,j of a determinant A we write in turn «„ + c, we get n' new determinants. If these be taken as the elements of another determinant its value will be where aS* is the sum of all the elements of A. 17. If u^{X^-afi:){X,-a,hy..{X„-aX), prove that the value of the determinant ^1' °-fii' "a^i •■• «»ii »i&3. aA' ^s ••• C'J'i «l^nJ tta^.I «A ••• ^n (l "l^. »,A ) 13 — 18.] EXAMPLES ON THE METHODS OF THE TEXT. 217 and the value of 0, »,, a, ... a„ 6,, X,, afi^ ... a„6, ^3, «,Ja. ^, ■■• «A 18 ^«. «!*».« A ■■• -^» 18. If u = {x-2a^ l^x-2a^) ... (x - 2a„), prove the following theorems : {x-ay, a/ , a^' ... ( X— 2aJ 0, 1,1, 1 1, (x-a;)', _ a,' , a/ 1, a/ , («-«,)", < 1, < , < , (x-a^)' „_,du 0, or, , a, , a, ... =-"-'4-^^^} 'a; -2a/ And if Z> = (a;-a,y, as/ ... a,' , 6,, 1 a," , a," ... (a; -«,)', 6., 1 h , K ■■■ K , 1 , 1 ,.. 1 218 then THEORY OF DETERMINANTS. «!"-'« \x-1a^ x-'hajXx-la^ '" x-2aj [ex. -t + ... + — l— ^ . x-2aj [x — 2a. 19. Prove that, if S = x + y + z+u, {S-uy, X' , y' , z' u' , {S-xY, y' , z' u' , ^ , (S-yY, ^ u' , x' , y' , (S-z) 0, 1 , 1 , 1, . 1 (11114) " (x y z u S) 1, {S-u) 1. u" 1, u' (S-xf, f x" , {S-yy rZ-¥y') + 2xyz + 2xzu + 2yzu + 2xyu -x'-y^-z'-u^ 20. If X = onxdnix, &c. prove that sno!, sn'iB, X sny, sn'y, F snz, Bn% ^ = sn (y — «) sn {z — as) sn (x-y)s,n{x + y + z) M, ■where -il^= 1 - ^ {sB^y SB?z + sv^z svlx + sv^x sn'?/} + lc'{\+¥) sn'a; sn'y sn'« - A'sn a; sn y sn » ( Z^ sn x + ZX sa.y+ Z7sn a). 21. If sn a; en a; dn a; = X, &c. prove that 1, sn'^a;, sn*a;, X =0, 1, sn'y, sn'y, F 1, sn'^a, sn*«, Z 1, sn^'w, sn^M, U provided x + y + z + u = 2^K + 2qiK', p, q being integers. 22. then If S. ^M-M> 'S't-ij.. 'S'»-»i- 18 — 24.] EXAMPLES ON THE METHODS OF THE TEXT. 219 is the sum of all the minors of order h-h of the determinant j1 = I ftjj I ; excepting always in such sum those determinants and their complements of order h which in their formation have two row or coliunn sufiixes congruent with regard to the modulus h. 23. If -D.= 0, 1, 1, 1, 1 ... (wrows), 1, 0, X, 0, 1, y, 0, X, 1, 0, y, 0, X 1, 0, 0, y, where all elements are zeros, with the exception of the border, and two lines of elements one on each side of the priacipal diagonal, prove that D^ -»2/An-l + x + y x"' + f_2(-xy)' x + y x + y and hence that -0^« = ^n+i ^ ^»»+i _ (2n + l)(x + y){-xyy 24. If D = (x + y)' h, c, a, c, c .. c, b, c, a, c .. c, c, b, c, a .. c, c, c, b, c .. (n rows), where all the elements are c with the exception of two lines, one on either side of the principal diagonal, prove that r («-c)"-(c-6)r ^»"-'"U a + 6-2c j Find also the value of D.,. 220 THEORY OF DETERMINANTS. [EX. 25. If Z>.= 0, 1, 1, 1, 1 ... (nrows), 1, c, a, 0, 1, b, c, a, 1, 0, 6, c, a 1, 0, 0, 6, c (where, with the exception of the border, the elements in the leading diagonal are a, in the lines on either side of it a and b, the rest are zero), then I>„-cI)„_, + abB„^,= _ (-«)"-+(-&)-• a + b + c c w"-' -v" -1 a+b + a u — V 2ab u"-' -v'- 3 a + b+ c u-v where u and v are the roots of the equation «" - ca + a6 = 0. Hence shew that u' + v' nc {u" + v") B. " (a + 6 + c)^ (a + 6 + c) (m - vf 2abn u"" + v"-' (-a)"+(-5)' a + b + c' {u — vf {a + b + cf 26. The value of the determinant "•i , ".J ... «„ M._,, U ...U „ M, , M, ... Wj (i) If M^ = a + (r - 1) 6 is 2a + (n-l)6. ,.._, (ii) If M, = a;'-' is (1-a;")'—. (iii) If w, = r' is 25 — 29.] EXAMPLES ON THE METHODS OF THE TEXT. 221 (iv) If u^ = cos {a + (r - 1) 6} is [cos a - cos (a + nh)]" - [cos (a-b) — cos {a+ (n—l) b}]' 2(1— cos»i6) (v) If M_.= siii{a + (r— 1)6} we must change the cosines in the numerator of (iv) into sines. (vi) If u, = »'-' + x'*"-"- + x'-"""-' + ... ad inf., is {l-xT- 27. The solution of the partial differential equation i)„ A ... i)„ M = 0, M = S-f (x^ — (DCCj , aSj — oi'x^ ... a;_, - T 3 a,a^...a^ And if 2 -g fa - «t)' ay, = cos(a, + a,. + a^), a,„=0, (- 1)-' Z> n-l + 2S cos 3a, cos 3^2 ■ ■ • cos 3a where i, k are all duads from 1, 2 . . . w. cos (a, + %) sin' (a, — a,) cos 3aj cos 3aj ' 31. If ^ = |ajj|, 5=|6„| are two determinants o£ orders n and m respectively, we can form a new square array of {mnf elements as follows. Repeat the array 6jj , n times in a row, and take n such rows, so that B is repeated Uke the squares on a chess-board. Then multiply each of the elements of that block which stands in the i*** row and A* column by a^. The determinant of the resulting array is equal to ^"-8". Example : A a, b c, d ; B = a, P 7. 8 aa, a/3, ay, aB, ca, c/3, cy, cS, 6a, bP by, bS da, dp dy, dS = !'£" 32. li a, b ... I ; u., p ...X are any two sets of n quantities, and d^ = (a. - a,y + {b, -p,y+... + (I, - x,y, prove that = 0, if « = »(r-l) + 3. < <> d 29 — 34.] EXAMPLES ON THE METHODS OF THE TEXT. 223 1 = 0, if «=M(r-l) + 2. 33. < In this and tlie next five questions _ m (ot - 1) (m — 2) . . . {m-k+l)'\ TO, The determinant (w+1). 1.2.3 ...k ""r+i ("»+l)p+. (m. + 1)„ (m+r-1), , (m + r-1),^, ... (m+r-l)„ (M + r), , (»» + »-)p4., ••• ("* + r)„ (m + r + «)p , (m + r + s),^, ... (TO + r+s)„ (m + r + s+l)„ (m+r+s + l),^„ ... (m + r + s+l)„ (?re + r + « + <)p, (?>i +»• + «+ <)p^, ... (m + r + s + «)„ where u=p+r + t + l (the suffixes ^, ^ + 1 ...wof the rows are con- secutive, but m, m + 1 ... m + r, m + r + s...m + r + s + t form two groups of consecutive numbers), is equal to the product of the two fractions m^(m + l\ ... {m, + r\(m + r + s)^ ... {m + r + s + t\ {r + s\^, (r + s + 1),^, ...{r + s + t\^^ (r+l),,.(r+2),,,...(r+« + l),,, " 34. The determinant "»„ m,^, ... m,+., m^^,^, ... ?7i,^.,^„^. {m+\\, (m + 1),^, ... {m+\)^^„ ("i+lW.-- ('«+lU+.^. {m+2\, (m + \,,... (m+2),,., (m + 2),,.,. . . . (m+2),,.,.,„ (m + r)„ (m + r),^, . . . (/rt + r\^„ (m + r\^,^, ...{m + r),^.^,^„ where r = s+u + \ (the suffixes p, p + \ ...p + s, p + s + v...p + s + v + u form two groups of consecutive numbers, while m, m+\ ...m+ r are consecutive), is equal to the product of the two fractions "t^ (m + 1) , ... {m + r\ i',(.p + 1), •••(;'+*),(;'+»+''),•■• i > r the deter- minant reduces to a function of x of order n — r. 36. Prove that X", p, ... p, (x+iy, {p + i\...{p+ii {X+2Y, {p + 2\ ... (p + 2l = (x-p)' (x+ry, (p + r)^ ... {p+rX for all positive values of n less than r. 37. Prove that P„ Pi ■■■ ;',-, n" =A'-n". (^ + 1)„, (;,+ l), ...(;, + 1X_., (n + iy {P + ^)o, (P + 2). -..(;' + 2X_„ {n + 2y {P + ^)o> (P + ^\ ■•• (P+'-X-i. (M + r)" 38. Prove that the value of the determinant (m-p)m^, nm^^^, gm^+i, tm^^^ {m-p + \){m+l\,{n+l)(m+l),^^, (?+l)(m+l),^„ (< + l)(m+l), (m-p + 2){m+2)„{n + 2){m+2),^^, {q+2){m+2)^^„ {t + 2){m+2)^. m„(m+l)- ... (m + r)- , ., ,, , ;\p^li...ip^r)^ '^-P^^^-P^')(^-P^')---('--P^^)' 13 and so is independent of the quantities n, q, t... 39. If ^ = I a,j I ; -S = I 6ji I are two determinants of order n, and f(x)= I a,, + xb,,\, prove that Aco)f{-x)^AB\H,-K,x''\, where the quantities ff^^, .^is,sat;i^fyth^ equations ^.^. + ^.^.+ - + ^„A'„=l, 35 — 41.] EXAMPLES ON THE METHODS OF THE TEXT. 225 40. With the same notation as in the preceding question, prove that if P(X, ^)= I Xa^ + fib^\, then Fi\,^) = A = B l^^u + K ^^12 ••• M-fii. XK^^ , \K,^ + ,. ... XA;„ A.A' 41. If F{x)-a^x" + a^a^~^ + ...+a,_^x + a^, prove that P = «, 0, ... 0, ^ - 1, .T, .. 0, '^ 0, -1, a; ... 0, a„ 0, 0, ... X, 0, 0, ...-1,^+5 ^ = 1 ! + -"-■ a;, -X, ... 0, F(x) 1, -£ 0, -^=x, 0, 1 ... 0, X, 0, ... 1, -X '-' X, 0, ... 0, 1 F{x) If P^^ , Q^^ be the coefficients of homologous elements in P and Q, a,P„x + a,^Q„ = F{x) a„P„x + o, = s.s__, + s_^, , w-2, -s, ... -s„_, -3. , ,n-3... -s„_^ , 0,0... 2 the number of positive integral solutions of the equation 44. If 8, is the sum of all the divisors of r, then the determinant rt-i rt-3 I, n-1, n-2, Sj , n-3 »„_3) o„_2 S»_lJ S„_3 S„_., S.._, «.-«3, , , ... 3, s, Si-s„ , , ... 0, 2 is equal to (- 1)* n ! when n is of the form J (Sic' ± /I), but vanishes for other values of n. 41 — 49.] EXAMPLES ON THE METHODS OF THE TEXT. 227 45. Let (m, n) denote the greatest common divisor of the integral numbers m and n ; and let xj/ (m) be the number of numbers not sur- passing m and prime to m ; the symmetrical determinant i)„=2±(l, 1)(2, 2)...(m, m) is equal to V.(1)^(2)^(3)...^H. 46. If ^ is a skew determinant of order n in which the principal diagonal elements are equal 'to z, and ^j^ its system of first minors, prove that A is equal to Aw„ if n is even, and to — w„ if n is odd. 47. If /(«)=«;" + a, a;""'+a,£c"-''+... + a„ = has for its roots b^, b^ ... b^, prove that /(^) = X, 6,, KK X, 6, ■•■ h^, b^ 1, 1, 1 ...1, 1 And if s^ is the sum of the r"" powers of the roots "jn-lJ °2»-2 ••• "nJ "n-l (-1) ^ tib,,b^...b„)f(x). 48. Prove that = t'' (a,, u.„ ... a) H ,,, ... a , 1 .Sy being the sum of the homogeneous powers and products of order p of a,, a ... a . 1' 2 n 49. If a , = , a„ = -. rj , prove that the value of the determinant of order 2n »i2. ".2---a„2. "„2 15—2 228 THEORY OF DETERMINANTS. [ex. where ^ (x) = {x- a;,) (x-x^) ...{x- xj. 50. Prove that the value of the determinant of order 2n+l whose 1, sinttj, cosa„ sin2a„ cos 2ai . . . sin »ia„ coswaj, IS S^'^nsinKa^-aj), where i, k are all duads from 1, 2 ... n{i>k). Also that the value of the determinant of order 2w whose i* row is sinaSj, coscSj, sin2aj, cos 2a( . . . sin Jia, , cos waS;, is where jS='%cos ^(a^ + a^+ ... + a^-a^^^ ... - a^J is formed by dividing the 2n angles into two sets of n in all possible ways and taking the cosine of half the difference of the sums of these sets. 51. If A = 1 1 1 ' a,,-x, 1 a,-x. ' a,-x. 1 1 a^~ x^ "l , 1 , 1 ■, 1 prove that ^ = (-!)' ^i(a,,a.^...aJ)^i{x^,x^...x^^J where i^ (x) = (x — xj (x — x^) ... (x — a:„+ 1). If B is the determinant obtained from A by writing (a^ — x)' in place of (a^ — x), prove that A B" ' 1 a,-x. 1 1 1 a^ — .Tj "l 1 1 , 1 , 1 \ 1 49 — 52.] EXAMPLES ON THE METHODS OF THE TEXT. 229 the function on the right being formed like a determinant, with all the signs positive instead of alternating. 52. If o, 13 ... \; a, j8' ... X' are two sets each of n quantities, and C, is the product of all the binomial coefficients in the expansion of (1 + x)', prove the following equalities : (a -a')", (a-;Q')"... (a-XO" (^-a')^ (|8-)8')"...(/3-X')" (X-a')", (\-^)-...(\-\')" where C = ^C4(a,/3...X)^i(a',^'...\')/, /= Ta — a', a — /3' ... a — X'1 j8-a', P-p'...p-X' If X-a', X-;8'...X-X' u = {x-ai/){x-Py) ... (x-ky), V = {x- a'y){x -P'y) ... {x~ k'y), /=(12)"My, using the notation of invariants, (a-a'Y ... (a-XY, (a-x)" (X-a)"... (X-X')", (X-a;)" (x-a'y ...{x-xr> {a-ay*' ...{a-X'y*\ (a-x)'^' = {-l)'C,^{a,p...X)^{a',P'...X')uv, C. (X-a')""... (X-X')"-", (X-a;)"+' (a:-a')"+' ... (aj-X')""^' where / = r a — a' . . . a — X', a — .v |8-a'.../3-X', P-x X^^(a',/3'...X')/.Mr, = -(12)"-'m«. X — a ... X — X , X — x .x — a ... x — X' Again, (a-a')"...(a-X')-, 1 (X-aT 1 (X-X')", 1 1 = (-l)"-(7.0(a...X)^(a'...X'), 230 THEORY OF DETEEMINANTS. (a-a')"+'...(a-X')"+', 1 [ex. (\-a')"+'... (X-X')"^', 1 1 ^i-'Tf^.^'i'^-^nH'^'-yy, r= [a-a! ...a-\', 1" 1x-a'...X-X', 1 L 1 1 = (12)"-'— — ' dx' dx' 53. Let there be two systems of binary n-tics u^ ... u^; v^ ...v^^ where M, = a,; x" + n^a^,x"-\j + np^^x'~Y + ... + a„,y", ^i = ^0, 3!" + '*i^iia'"~'2/ + '»2^2i«!"~''2/^ + . . . + 6„,.3(°. And let (i, h) be the lineo-linear invariant of u^ and v^, so that Prove that (1,1) ... (1,^+2) (71+2, l)...(9l+2,TO + 2) = 0, (1,1) ... {l,n + \) {n + l, 1). ..(«.+ 1,M+ 1) = «„,, «!, ... a„i '"'on+D^^ln+l ••• ''^»n+l 54. If a,, a^ ... a^ are the roots of the equation X +PrC prove that 55. If ■+i'„=0. l«, = — . . . M„ , = ^i- a;^ being a function of a;,, x^... ^c„_^ given by a;,'' + a;/+...+a;„_,= + a!/=l, prove that d{x^, X,... x„_,) «:+'■ 62 — 59.] EXAMPLES ON THE METHODS OF THE TEXT. 231 56. If u^ = {x + y + z)'' + ix-y-z)" + {-x + y-zY + {-x-y + zY, prove that the Hessian of m„ is M^_„ {x" +y* + z*- IxY - S^/"*' - 23'a!')"-'' multiplied by a numerical factor. 57. If F = U,U„ ... M„ •where v^, zt^ ■■•«„ are linear functions of the n variables x^, x^...x^, prove that ™(Iog^) = (-l,-[^|^i^J]-. Also that dF dx^ = (_l)»i?"-' 'd{u^ . [d {x^ . ..x„) dF d'F d^F dx^' dx^ dx^dx^ dF d'F d'F dxj dx^dx^ ' ■ d<«n 58. If Mj, Mg, Mg be three functions of a?, y, and if d (m„ M3) c?(m„mJ yfa,,zt^) . d(v„V s) "'1- prove that Wi= y, "' "' , &c., 59. If w,, t4j, M3, M^ are four functions of a;, y, and if v.= (Fu^ d\ d\ dx' ' dx' ' day' d^u, d'u, d^u. dxdy ' dxdy' dxdy d\ d\ d^u^ 'df' w W and Vj, v^, v^ similar determinants formed from u^, u^, m_, &c., then 232 THEORY OF DETERMINANTS. [ex. from V,, v^, v^, v^ we can form four new functions w,, w^, w^, w^ in the same way as we obtained v^ ... ■y^ from u^ ...u^. Prove that -=M d\ d\ d\ d\ dx" dx" dx'' dx' d'u d'u^ d^u. (fu dafdy' dxfdy' dx'dy' dyfdy d^u^ d?u^ d^u^ d^u^ dxdy^ ' dxdy^' dxdi/" dxd'if d'u^ d^u^ d'u^ d^u^ ~d^' 'df' ~df' H^ where /* is a numerical factor. 60. For the n' functions Mjj {i, k=\, 2 ... n) of the Tariables ajj, x^ ... x^, prove that the cubic determinant whose elements are du,. dx, ^' {i,3,k=\,'l...n) is a covariant. 61. For the n functions m^ . . . m„ of the variables x^...x„, prove that the cubic determinant whose elements are d^u^ dx.dx^ {i,j,k=l,2...n) IS a covariant. 62. If the function u of the variables a;, . . . a;„ be transformed by the linear substitution aJ. = 5„2/, + 6„y, + . . . + 6,„.,y„.. to a function «; of w - 1 variables, prove that 0, A • ..£. ^.= w„ . •■■^M ^„ «„. • • M™ where u^= , . , and (— 1)'-B( is the determinant obtained by suppress- ing the i"" row in the array formed by the quantities 6, . 63. If w= SttiiajjiCj (i, Z; = 1, 2 ... n), and J) 59 — 66.] EXAMPLES ON THE METHODS OF THE TEXT. 233 prove that the substitution K=y,-^ 1 dD.. 1 dD ■2/r+l+-+7) D da ^/'+' D^ ,da •'" r rr+ 1 n—\ rn reduces tlie giyen quadric to the sum of the n squares u = %^y: (r= 1,2. ..,»). r—\ 64. If w and v are two w-ary quadrics and U, V their reciprocals, prove that -we can by the same linear substitution change m into A V and V into BW; A and B are the discriminants of u and v. The determinant C of the substitution is the geometric mean between the discriminants of U and F. If C be regarded as the discriminant of a quadric W, we can by the same linear substitution reduce the three quadrics U, V, W to the sum of squares. The coefficient of any term in W so transformed is the geometric mean between the homologous co- efficients in U and F. 65. If to the leading elements of the determinant of an orthogonal substitution of order n we add the quantities a^^, a^ ... a^, or the quantities — , — . . . — , the resulting determinants are equal if 66. If Cj^are the coefficients of an orthogonal substitution (modulus unity) of order n, prove that i> = 1 ... '^nl > "nil ••• "nn ^ is equal to zero if n is oddj but if n is even its value is ^ "X- •where A is the skew determinant from which the orthogonal substitution is derived, and [AJ the same determinant with the elements in the lead- ing diagonal zero. If Z>„ is the coefficient of one of the leading terms in B, prove that when n is even 2D, = _i). 234 THEORY OF DETERMINANTS. [ex. 67. If |cj = £ is the determinant of an orthogonal substitution, the equation "nl ) "nS c^. + x = is a reciprocal one. li nis odd it has one real root — e ; if w is even and e = - 1 it has the two real roots ± 1. The rest are all imaginary. 68. The maxima and minima values of subject to the conditions c„_,,a;,+c„_^,a;^+... + c„_,„a3„=0 are given by the equation b^^u-a^^v ... 6i„M -»,„«, c„, c,, ... c„_,, &„iM-a„i« ••• K.u-a„„v, Ci„, C2„ ... c„ 69. The values of x,, a;^ ... a;^ which satisfy the equations a..x + o a;„ + ... + a„,a;„=0 111 SI 2 nil m a,, a;, + a„„a3„ + . . . + a a;„ =0 and make x," + a;/ + . . . + a; J a minimum are where C is the determinant whose elements are given by 67 — 72.] EXAMPLES ON THE METHODS OF THE TEXT. 235 70. The value of the integral II ... x^Xjdx^dx^ . . . dx^, taken for all values of the variables such that the quadric being a definite positive form (i.e. incapable of becoming negative), is where J. = | »,j | is the discriminant of the quadric. 71. The value of the integral I ... I i~''coa(b^x^ + b^x^+ ... +b„x^)dXjdx^ ... dx^, J —00 J —03 u=-$a,,x.x„ ■where IS where ^ = __ 0, 5., 6, ... b„ 5„. «nl) »»2 ■-•», In this question and the next u is supposed to be incapable of becoming negative. 72. The value of the integral I ... I ve~"dx^dx^ ... dx^, J -co J-ao V = %b^^XiX^, u = %a^^x^x„ where IS where S is the sum of the n determinants obtained by substituting for each column of A in succession the corresponding column of the dis- criminant of V. 236 THEORY OP DETERMINANTS. [ex. 73. Let a,, a^ ... a^^+i ^^ 2n+l real and different numbers in ascending order of magnitude, and let P{x) = {x-a^){x-a^) ... (x-a^^,) Q{x} = (x-as)(x-a^) ... {x-a,^)A B{x)=.P{x)Q{x), A being a positive number. Then if r«^ P{x)dx _ Q{a,^_,) /•«!. P{x)dx " ia,_S^~<^.r-.)slR{^)' " ^P\a,^_,)j^^Jx-a,_,YJR{x) (these are the complete Abelian integrals of the first and second species), and if also , ^vQ {""^ ^ ^ ('") ^=^ 7 =n/z1 C =(!)■ ProTe also that dP dB _ _/^Y"', c^i) <;Z) c?i;„ £^x.. -©■ ^„-i dD_dD^ dk,. dk _ (J\-' J dB dB /^V-\ 74. Prove that the value of the continued fraction a h c a+l — b + l— c + 1- ad. inf. is unity. 75. Prove that the product of the two continued fractions a-l + a+l + 2{a-iy+ 2(a-iy+'-- V 3' 2(0+1)''+ 2{a+iy+ IS o . 73 — 80.] EXAMPLES ON THE METHODS OP THE TEXT. 237 76. If M„ is the number of terms in a determinant of order n ■which, do not contain any element from tlie principal diagonal, prove that w„ = iM„.,+ (-l)", and hence that -^, is the coefficient of x' in the expansion of :; . n\ ^ \-x 77. If w„ is the number of terms in a symmetrical determinant of order n, prove that (»-l)(w-2) M,. - '»M»-i + -^ «*„-s= 0- Also that —. is the coefficient of x" in the expansion of Ax+\x'' 78. If [1.3.5... ip,n— 1)] M„ is the number of terms in a skew determinant of order In, prove that M„= (2w - 1) «„., - {n - 1) M„.,. Shew also that -^^r-. is the coefficient of x" in the expansion of vli-J- 79. If A is the area of a quadrilateral, the co-ordinates of whose angular points are (x^, y^ ... (x^, yj, then 1, 0, «x, y, 2A 0. 1> ^„ y, 1. 0; 333, y^ a'«-a'2> 2/. -2/2 0, 1, «„ y, The area of a quadrilateral inscribed ia a circle in terms of its sides is given by 16^ = - -a, h , c , d b , — a, d , c G , d , —a, b d , c , b , -a 80. If the planes a^x + b-y + CiZ + d^=0 (i = 1, 2, 3, 4, 5) touch the same sphere, then I »i, *(> c„ d„ mJ = (i=l, 2 ... 5), where u' = a.' + b'' + c'. 238 THEORY OF DETERMINANTS. [ex. 81. A quadric of revolution passes tlirougli five points P^...Py The distances of these points from a focus being n ... r^. If Fi = volume of tetrahedron F^ P^ P^ P^, he, prove that F,r,+ F,r,+ ...+ F,r^ = 0. 82. Let F, F' be the volumes, A, B, G, D; a, h, c, d the areas of the faces of two tetrahedra whose angular points are numbered 1, 2, 3, 4. Also let Pa be the perpendicular from the point i of the first tetrahedron on the face opposite the point h of the second, and p^ a like quantity for the other tetrahedron. Prove that x|P« {VV'Y ■ ' ABGDabcd (i, ^ = 1,2, 3, 4). 83. If A, B, C, B are the directions of four forces in equilibrium, and if AB is the moment of the Unes A and B, &c., prove that , BA, CA, DA =0. AB, , CB, DB AC, BC, , DC AD, BD, CD, If a, b, c, d are the magnitudes of the forces a = J{BC.CD.DB), &c. 84. In Siebeck's determinant, xiv. 22, prove that ^ = 288..', dd^ where v is the volume of the tetrahedron formed by the face opposite the point i of the first tetrahedron and the centre of the sphere circum- scribing the second tetrahedron, and similarly for v'. 85. If in a system of five points d^ is the square of the line joining the i* and k"' points, and r is a sixth point of the system, prove . ,. .. .. =0. d.,d_+d„ that drA6 + <^n, d^A^+d,,... d^, d^, + l d,+ l d.,+ l 5+1 1 81 — 88.] EXAMPLES ON THE METHODS OF THE TEXT. 239 86. If in a system of seven straight lines, m^ is tlie moment of the t" and A"" lines, and r is an eighth line, prove that .m_,m.,+ m,. = 0. m^j m,, + m,„ m^ m^, + m^, . . . mi^/ 87. Having_ given two tetrahedra whose angular points are marked 1, 2, 3, 4, let cl^^ denote the square of the distance between the i"" point of the first and A" point of the second tetrahedron. Prove the following relations : (i) For two points P, Q the distances of F from the angular points of the first tetrahedron being a., of Q from those of the second 6„ and = 0. d, 1, ^ • ■K 1- 0, 1 . .. 1 5^1,- 1, ^u- ..d. at, 1, d^^...d^, (ii) For the point P and a plane, q, being the distances of the vertices of the second tetrahedron from the plane, p the distance of P from the plane, p> 0, y. ■ ■ 1^ 1, 0, 1 . . 1 «1. 1, rf.,. < «4. 1, d^. ■< (iii) For two planes, p , q being the perpendiculars from the angular points of the tetrahedra on them, ^ the angle between the planes. ^cos^, 0, q, ... qt , 0, 1 ... 1 Pi , 1, dii...d,, P4 1, rf„ ... du = 0. 88. For a system of six and a second system of five spheres, if p^ is the power of the i"" and A" spheres, 1, Pu ■■■Pis =0- 1, P6i---P,s 240 THEORY OF DETERMINANTS. 89. The equation = [ex. J ^l> ^2! ^3' ^i 'S'l, 0, «,3, «i3, {x, y, z), r,=x^{x, y, z), ^ = x(a!, y, a), prove that the ratio of corresponding elements of the surfaces is given by da- d^ di dt ds dx' dy' dz' a. df) d^' dy) dy' drj d^' P di dl_ dK dx' dy' dz' "^ a , b , c where (a, b, c), (a, j8, y) are the direction cosines of the normal to ds and da: S. D. 16 LIST OF MEMOIRS AND WORKS RELATING TO DETERMINANTS. Albeggiani, M. Sviluppo di un determinante ad elementi binomii. Giomale, x. 279. (1872.) Sviluppo di un determinante ad elementi polinomi. Ibid. XIII. 1. (1875.) ■ ■ Dimostrazione d' una formula d' analisi di F. Lucas. Ibid. 107. (1875.) Armenante, A. Sui determinanti cubici. Giomale, vi. 175. (1866.) Baltzer, R. Ueber den Ausdruck des Tetraeders durcb die Coordina- ten der Eckpuncte. Crelle, lxxiii. 94. (1870.) — — — — Theorie und Anwendung der Determinanten. Ed. 4, Leip- zig. (1875.) Battaglini, G. Sull' equUibrio di quatro forze nello spazio. Giomale, IV. 93. (1866.) Nota sui determinanti. Ibid. ix. 136. (1871.) Bauer, G. Bemerkungen iiber einige Determinanten geometriscber Bedeutung. Sitzungsber. d. k. baierischen Akad. zu Mimchen, ll. 345. (1872.) Baur, C. W. Auflbsung eines Systems von Gleichungen, worunter eine quadratiscb, die anderen linear. Zeitschrift fur Math. u. Physik, XIV. 129. (1868.) Bazin, M. Sur une question relative aux determinants. Lioumlle, XVI. 145. (1851.) ■ Demonstration d'un theorfeme sur les determinants. Ibid. XIX. 209. (1854.) Becker, J. C. Ueber ein Fundamentalsatz der Determinantentheorie. Zeitschrift, xvi. 526. (1871.) Bertrand, J. M6moire sur le determinant d'un systfeme des fonctions. Liouv. XVI. 212. (1851.) LIST OF MEMOIRS AND WORKS. 243 Bjorling, C. F. C. Sur les relations qui doivent exister entre les coefficients d'un poljndme pour qu'il contienne un facteur de la forme, (a;" - a"). Grunerts Archiv, hv. 4:29. (1873.) Bonolis, A. Sviluppi di alcuni determinanti. Giornale, xv. 113. (1877.) Brill, A. TJeber zwei Beruhrungs-Probleme. Math. Ann. iv. 527. (1871.) Brioschi, F. Sur une propriety d'un produit de facteurs lingaires. Camb. (Sc Dublin M. J. ix. 137. (1853.) Sur quelques questions de la geometrie de position. Crelle, L. 233. (1854.) Note sur un theorfeme relatif aux determinants gauches. Liouv. XIX. 253. (1854.) ■ Sur deux formules relatives k la theorie de la decomposition des fractions rationelles. Crelle, l. 239 and 318. (1855.) ■ Sur I'analogie entre une classe de determinants d'ordre pairs; et sur les determinants binaires. Crelle, lii. 133. (1855.) Sur une propriete d'un determinant fonctionnel. Quart. J. I. 365. (1855.) Theorie de determinants. (French trans, by Combescure) Paris, (1856.) Sur une nouvelle propriete du resultant de deux Equations alg^briques. Crelle, Liii. 372. (1856.) Sullo sviluppo di un determinante. Annali, i. 9. (1857.) SuUe funzioni Abeliane complete di prima e seconda specie. Ibid. 12. (1857.) Intorno ad un teorema del Sis. Borcbardt. Ibid. I. 43. (1857.) Sulle funzioni Bernouilliane ed Euleriane. Ibid. 260. (1857.) Intorno ad una trasformazione delle forme quadratiche. Giornale, I. 26. (1863.) Bruno, Faa de. Demonstration d'un tbeorfeme de M. Sylvester relatif k la decomposition d'un produit de deux determinants. Liouv. XVII. 190. (1852.) Note sur un tbeoreme de M. Brioschi. Ibid. xix. 304. (1854.) Calderera, F. Nota su talune propriety dei determinanti, in ispecie di quelli a matrici composte con la serie dei numeri figurati. Giornale, ix. 223. (1871.) Casorati, F. Intorno ad alcuni punti della teoria dei minimi quadrati. Annali, I. 329. (1858.) Sui determinanti di funzioni. Mem. del. reale Institiito Lomhardo di Scienze e Letters, xiii. 181. (1874.) 244 THEORY OF DETERMINANTS. Cayley, A. On the theory of determinants. Camh. Phil. Trans. VIII. 75. (1843.) Sur quelques proprietes des determinants gauches. Grelle, XXXII. 119. (1846). Sur les determkiants gauches. Ibid, xxxviii. 93. (1847.) On a theorem in the geometry of position. Cambr. M. J. II. 267. (1847.) On the rationalisation of certain algebraical equations. Camh. 6a Bub. M. J. viii. 97. (1853.) Eecherches ulterieures sur les determinants gauches. Crelle, L. 299. (1854.) Theorfeme sur les determinants gauches. Crelle, lv. 277. (1857.) On the number of distinct terms in a symmetrical or par- tially symmetrical determinant. Monthly Notices of R. Ast. Soc. XXXIV. 303, 335. (1874.) Note on a theorem in determinants. Quart. J. xv. 55. (1878.) Christoffel, E. B. TJeber die lineare Abhangigkeit von Fuuctionen eiaer einzigen Yeranderlichen. Crelle, lv. 281. (1857.) Clebsch, A. Ueber eine Eigenschaft von Functional-Determinanten. Grelle, LXix. 355, lxx. 175. (1868.) ClifTord, W. K. Analytical Metrics. Q. J. vii. 54. (1866.) viii. 16 & 119. (1867.) Crocchi, C. Sopra le funzioni Aleph ed il determinante di Cauchy. Giornale, xvii. 218. (1879.) Cunningham, A. An investigation of the number of constituents, elements and minors of a determinant. Quart. J. of Science. (1874.) 212. Darboux, G. Sur les relations entre les groupes de points de cercles et de spheres dans le plan et dans I'espace. Ann. Scientifiques de VEc. Normale Sup. Ser. 2, i. 323. (1872.) ■ • Memoire sur la theorie algebrique des formes quadratiques. Liouv. XIX. Ser. 2. 347. (1874.) Dickson, J. D. H. Discussion of two double series arising from the number of terms in determinants of certain forms. Proc. Math. Soc. X. 120. (1879.) Dietrich, M. Zur Theorie der Determinanten. Grunert, XLiv. 344. (1865.) Ueber den Zusammenhang gewisser Determinanten mit Bruchfunctionen. Crelle, lxix. 190. (1868.) LIST OF MEMOIRS AND WORKS. 245 Donkin, W^. F. Demonstration of a theorem of Jacobi relative to functional determinants. Gamb. & Duhl. M. J. ix. 161. (1854.) Dostor, G. Propriete des determinants. Grunert, lti. 238. (1874.) ■ Surface des quadrilatferes exprimfie en determinants. Ihid. 240. Le tri^dre et le tStrafedre, avec application des determinants. Ibid. Lvii. 113. (1875.) ifilements de la tteorie des determinants. Paris. 1877. D' Ovidio, E, Due teoremi di determinants Giornale, I. 135. (1863.) Alcune relazioni fra le mutue distanze di pii punti. Ibid. IX. 211. (1871.) Nota sui determinanti di determinanti. Atti della Reale Accad. d. Scienze di Torino, xi. 949. (1876.) Enneper, A, Zur Tteorie der bestimmten Integrale. Zeitschrift, vi. 289. (1861.) TJeber eine Determinante bestimmter Integrale. Ibid. xi. 69. (1866.) Eugenio, V. Considerazioni intorno a taluni determinanti particolari. Giornale, viii. 285. (1870.) Ferrers, N. M. On certain properties of the tetrahedron. Quart. J. V. 167. (1861.) On tetrahedi'al and quadriplanar coordinates. Ibid. 172. Fiore, V. Dimostrazione d' una trasformazioae di determinanti. Giornale, x. 170. (1871.) Fontebasso, D. I primi elementi della teoria dei determinanti e loro applicazioni air algebra ed alia geometria. Treviso. 1873. Franke, E. TJeber Determinanten aus Unterdeterminanten. Crelle, LXi. 350. (1862.) Frobenius, G. TJeber die Determinante mehrerer Functionen einer Variablen. Crelle, lxxvii. 245. (1873.) Anwendung der Determinanten-Theorie auf die Geometric des Maasses. Crelle, lxxix. 185. (1874.) Ueber das Pfaff'sche Problem. Crelle, lxxxii. 230. (1876.) Furstenau, E, Darstellung der reellen "Wurzeln algebraischer Glei- chungen durch Determinanten der CoeflScienten. Marburg. 1860. There is a review of this work by Baltzer, Zeitschrift, vi. Literatur- zeitung, 69. ■ Neue Methode zur Darstellung und Berechnung der imagi- naren Wurzeln algebraischer Gleichungen durch Determinanten der Coefficienten. Marburg. 1867. TJeber Kettenbriiche hoherer Ordnung. Programm des kgl. 246 THEORY OF DETERMINANTS. Realgymnasiums zu Wiesbaden. (1872.) Reviewed by Giinther, Lit. Bericht. Grunert, lvii. Garbieri, G. I determinanti con numerose applicazioni. Part I. Bologna. 1874. ■ Determinanti formati di elementi con un numero qualunque d' indici. Giornale, xv. 89. (1877.) Nuovo teorema algebrico e sua speciale applicazione ad una maniera di studiare le curve razionali. Giornale, xvi. 1. (1878.) Gasparis, A. de. Sopra due teoremi dei determinanti a tre indici ed un altra maniera di formazione degli elementi di un determinante ad 7n indice. Rendiconti delV accad. delle scienze fis. e. math, di Napoli, VII. 118. (1868.) Glaisher, J. W. L. On the problem of the eight queens. Phil. Mag. 457. (1874.) Theorem relating to the differentiation of a symmetrical determinant. Quart. J. xiv. 245. (1877.) On the factors of a special form of determinant. Ibid. xv. 347. (1878.) Expressions for Laplace's coefficients, Bernouillian and Eu- lerian numbers &c. as determinants. Mess, of Math. vi. 49. (1877.) • On a class of determinants, vii. 160. viii. 158. On a special form of determinant and on certain functions of n variables analogous to the sine and cosine. Quart. J. xvi. (1878.) Grundelfinger, S. Ueber einen Satz aus der Determinanten-Theorie. Zeitschrift, xviii. 312. (1873.) Aufldsung eines Systems von Gleichungen, worunter zwei quadratisch und die iibrigen linear. Ihid. 543. Gubba, A. Esposizione del principio d' elasticitiY e studj su talune sue applicazioni mediante e determinante. Mem. d. Reale Istituto Lomhardo di Scienze e Lettere, xill. 105. (1874.) Guldberg, A. S. Determinantemes teori. Kristiania. 1876. Gunther, S. Beitrage zur Theorie der Kettenbriiche. Grunert, lv. 392. (1873.) Ueber einige Determinantensatze. Sitzungsher. d. phys. med. SoG. zu Erlangen. (1873.) 88. Darstellung der JSTaherungswerthe von Kettenbriichen in independenter Form. Erlangen. 1873. Zur mathematischen Theorie des Schachbretts. Grunert, LVi. 281. (1874.) Ueber aufsteigende Kettenbruche. Zeitschrift, xxi. 178. (1876.) LIST OF MEMOIRS AND WORKS. 247 Gunther, S. Das allgemeine Zerlegungsproblem der Determinanten. Grunert, lix. 130. (1876.) Lehrbuch der Determinanten-Theorie. 2nd ed. Erlangen. 1877. Von der expliciten Darstellung der regularen Determinan- ten aus Binomialcoefficienten. Zeitschrift, xxiv. 96. (1879.) Eine Relation zwischen Potenzen und Determinanten. lUd. 244. Gyergyoszentmiklos, D. D. de. Resolution des systfemes de con- gruences lineaires. Gomptes Rendus, Lxxxviii. 1311. (1879.) Hankel, H. Ueber eine besondere Classe der symmetrischen Deter- minanten. (Inaug. Diss.) Gottingen. 1861. Ueber die Transformation von Reihen in Kettenbruche. Zeitschrift, vii. 338. (1862.) Also, Sitzungsber. d. hgl. Sachs. Ges. d. Wissenschaften. (1862.) Vorlesungen uber die complexen Zatlen und ihre Functionen. Leipzig. 1867. Hesse, O. Ueber Determinanten und ihre Anwendung in der Geome- trie, insbesondere auf Curven vierter Ordnung. Grelle, xlix. 243. (1853.) Ein Determinantensatz. Grelle, lxix. 319. (1868.) ■ Ein Cyclus von Detarminantengleicbungen. Grelle, lxxv. 1. (1872.) Die Determinanten elementar bebandelt. Leipzig. 1872. Horner, J. Notes on Determinants. Quart. J. viii. 157. (1865.) Hoza, F. Beitrag zur Theorie der Unterdeterminanten. Grunert, LIX. 387. (1876.) Ueber Unterdeterminanten einer adjungirten Determinan- te. Ibid. 401. (1876.) Hunyady, E. v. Ueber ein Produkt zweier Determinanten. Zeit- schrift, XI. 359. (1866.) . Ueber zwei geometrische Probleme. Ibid. 64. Ueber Yolumina von Tetraedern. Ihid. 163. • Ueber einige Identitaten. Zeitschrift, xii. 89. (1867.) Jacobi, G. C. J. De formatione et proprietatibus deter minantium. Grelle, xxii. 285. (1841.) De determinantibus functionalibus. Ibid. 319. . De functionibus alternantibus earumque divisione per pro- ductum e differentiis elementorum conflatum. Ibid. 360. Janni, G. Nota sopra i determinanti minori di un dato determinante. Giomale, I. 270. (1863.) 248 THEORY OF DETERMINANTS. Janni, G. Teorica di determmanti simmetrici gobbi. Ihid. 275, Janni, V. Sul grado dell' eliminante del sistema di due equazioni. Ibid. XII. 27. (1874.) Dimostrazione di alcuni teoremi sui determinanti. Ibid. 142. Joachimsthal, F. Sur quelques applications des determinants k la geometrie. Crelle, XL. 21. (1849.) Note relative k un tbeoreme de M. Malmsten sur les equa- tions diiferentielles lineaires. Ihid. 48. De aequationibus quarti et sexti gradus quae in theoria linearum et superficierum secundi gradus ocourrunt. Crelle, Llll. 149. (1856.) Kostka. TJeber ein bestiinmtes Integral. Zeitschrift, xxii. 258. (1877.) Kronecker, L. Bemerkungen zur Determinanten-Theorie. Crelle, Lxxii. 152. (1869.) Mansion, P. Elements de la tbeorie des determinants. Mons. 1875. On an arithmetical theorem of Prof. Smith's. Mess, of Math. Ser. 2. vii. 81. (1878.) On rational functional determinants. Ibid. IX. 30. (1879.) Matzka, W. Grundzuge der systematischen Einfiihrung und Be- grundung der Lehre der Determinanten. Abh. d. k. bohm. Ges. d. Wiss. IX. (1877.) ' Mertens, F. Anwendung der Determinanten in der Geometrie. Crelle, Lxxvii. 102. (1873.) TJeber die Determinanten deren correspondirende Elemente ttpg und a,p entgegengesetzt gleich sind. Crelle, Lxxxii. 207. (1876.) Nagelsbach, H. TJeber die Resultante zweier ganzen Functionen. Zeitschrift, xvii. 333. (1872.) Zur independenten Darstellung der Bemouilli'schen Zah- len. Ibid. xix. 219. (1874.) Studien zu Fiirstenau's neuer Methode der Darstellung und Berechnung der Wurzeln algebraischer Gleichungen durch Deter- minanten der Coefficienten. Grunert, Lix. 147. (1876.) Neumann, C. Zur Theorie der Functional-Determinanten. Math. Ann. I. 208. (1869.) TJeber correspondirende Flachenelemente. Ibid. xi. 306. (1877.) Padova, E. Sui determinanti cubici. Giornale, vi. 182. (1868.) Pasch. Note tiber die Determinanten welche aus Functionen und deren Differentialen gebildet werden. Crelle, lxxx. 177. (1874.) Picquet. MImoire sur I'application du calcul des combinaisons a la LIST OF MEMOIRS AND WORKS. 249 throne des determinants. Journ. de VEcole Polyt. xxviii. 201. (1878.) Pokorny, M. Determinanty a vyssi rovnice. V Praze. (1865.) Puchta, A. Ein Determinantensatz u. seine Umkelirung. Denh- schriften d. k. Akad. d. Wiss. Wien. (1878.) Reiss, M. Beifcrage ziir Theoiie der Determinanten. Leipzig. 1867. Ritsert, E. Die Herleitung der Determinante fiir den Inhalt des Dreieckes aus den drei Seiten. Zeitschrift, xvii. 518. (1872.) Roberts, S. Note on certain Determinants connected with algebraical expressions having the same form as their component factors. Mess, of Math. viii. 138. (1879.) Rosanes. Ueber Functionen, welche ein den Functional-Determinan- ten analoges Yerhalten zeigen. Crelle, lxxv. 166. (1872.) Rouche. Sur les fonctions X^ de Legendre. Comptes Bendus, xlvii. 919. Rubini, R. Su talune formole relative a determinanti. Giornale, iv. 187 & Rendiconto delV Accad. di Napoli, v. 109. (1866.) Salmon, G. On the relation which connects the miitual distances of five points in space. Quart. J. iii, 202. (1859.) • Lessons in higher algebra. Ed. 3. Dublin. 1876. 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PUBLICATIONS OF THE BACCHAE OF EURIPIDES, with Introduction, Critical Notes, and Archaeological Illustrations, by J. E. Sandys, M.A., Fellow and Tutor of St John's College, Cam- bridge, and Public Orator. \Nearly ready, PINDAR. OLYMPIAN AND PYTHIAN ODES. With Notes Explanatory and Critical, Introductions and Introductory Essays. Edited by C. A. M. Fennell, M.A., late Fellow of Jesus College. Crown Oc- tavo, cloth. 9^". "Mr Fennell deserves the thanks of all classical students for his careful and scholarly- edition of the Olympian and Pythian odes. He brings to his task the necessary enthu- «;iasm for his author, great industry, a sound judgment, and, in particular, copious and minute learning in comparative philology. To his qualifications in this last respect every page bears witness." — A thetKennt. "Considered simply as a contribution to the study and criticism of Pindar, Mr Fen- nell's edition is a work of great merit. But it has a wider interest, as exemplifjdng the change which has come over the methods and aims of Cambridge scholarship within the last ten or twelve years. . . . The short introductions and arguments to the Odes, which for so discursive an author as Pindar are all but a necessity, are both careful and acute. . . Altogether, this edition is a welcome and wholesome sign of the vitality and de- velopment of Cambridge scholarship, and we are glad to see that it is to be continued." — Saturday Retieix}. THE NEMEAN AND ISTHMIAN ODES. [Preparing, PLATO'S PH^DO, literally translated, by the late E. M. Cope, Fellow of Trinity College, Cambridge. Demy Odlavo. 5J. ARISTOTLE. THE RHETORIC. With a Commentary by the late E. M. COPE, Fellow of Trinity College, Cambridge, revised and edited for the Syndics of the University Press by J. E. Sandys, M.A., Fellow and Tutor of St John's College, Cambridge, and Public Orator. With a biographical Memoir by H. A. J. MUNRO, M.A. Three Volumes, Demy Oflavo. £1. i is. 6d. "This work is in many ways creditable to the University of Cambridge. The solid and extensive erudition of Mr Cope himself bears none the less speaking evidence to the value of the tradition which he continued, if it is not equally accompanied by those qualities of speculative originality and independent judg- ment which belong more to the individual writer than to his school. And while it must ever be regretted that a work so laborious should not have received the last touches of its author, the wannest admiration is due to Mr Sandys, for the manly, unselfish, and un- flinching spirit in which he has performed his most difficult and delicate task. If an English student wishes to have a full conception of what is contained in the Rhetoric of Aris- totle, to Mr Cope's edition he must go." — Acade7ny. "Mr Sandys has performed his arduous duties with marked ability and admirable tact. ...Besides the revision of Mr Cope's material already referred to in his own words, Mr Sandys has thrown in many useful notes; none more useful than those that bring the Commentary up to the latest scholarship by reference to important works that have ap- peared since Mr Cope's illness put a period to his labours. When the original Com- mentary stops abruptly thrse chapters be- fore the end of the third book, Mr Sandys carefully supplies 'the deficiency, following Mr Cope's general plan and the slightest available indications of his intended treat- ment. In Appendices he has reprinted from classical journals several articles of Mr Cope's ; and, what is better, he has given the best of the late Mr Shilleto's 'Adversaria.' In every part of his work — revising, supple- menting, and completing — he has done ex- ceedingly well." — Exatnifier. ** A careful examination of the work shows that the high expectations of classical stu- dents will not be disappointed. Mr Cope's * wide and minute acquaintance with all the Aristotelian writings,' to which Mr Sandys justly bears testimony, his thorough know- ledge of the important contributions of mo- dern German scholars, his ripe and accurate scholarship, and above all, that sound judg- ment and never-failing good sense which are the crowning merit of our best English edi- tions of the Classics, all combine to make this one of the most valuable additions to the knowledge of Greek literature which we have had for many years." — Spectator. _ * ' Von der Rhetorik ist eine neue Ausgabe mit sehr ausfuhrlichem Commentar erschie- nen. Derselbe enthalt viel schatzbares .... Der Herausgeber verdient fur seine miihe- voUe Arbeit unseren lebhaften Dank."— Susemikl in Bursian^s Jakresbericht. London : Cambridge Warehouse^ 1 7 Paternoster Row. THE CAMBRIDGE UNIVERSITY PRESS. ii P. VERGILI MARONIS OPERA cum Prolegomenis et Commentario Critico pro Syndicis Preli Academici edidit Benjamin Hall Kennedy, S.T. P., Graecae Linguae Professor Regius. Extra Fcap. Odlavo, cloth. 5J-. M. T. CICERONIS DE OFFICIIS LIBRI TRES, with Marginal Analysis, an English Commentary, and copious Indices, by H. A. HOLDEN, LL.D. Head Master of Ipswich School, late Fellow of Trinity College, Cambridge, Classical Examiner to the University of London. Third Edition. Revised and considerably enlarged. Crown Ocflavo. <^s. "Dr Holden truly states that 'Text, index of twenty-four pages makes it easy to Analysis, and Commentary in this third edi- use the book as a storehouse of information tion have been again subjected to a thorough on points of grammar, history, and philo- revision.' It is now certainly the best edition sophy. . . . This edition of the Offices, Mr extant. A sufficient apparatus of various Reid's Academics, Laslius, and Cato, with readings is placed under the text, and a very the forthcoming Pitt Press editions of the De careful summary in the margin. The Intro- Finibtts and the De Natura Deorum will do duction (after Heine) and notes leave nothing much to maintain the study of Cicero's philo- _to be desired in point of fulness, accuracy, sophy in Roger Ascham's university." — Notes 'and neatness; the typographical execution tutd Queries. will satisfy the most fastidious eye. A careful M. TULLII CICERONIS DE NATURA DEORUM Libri Tres, with Introduction and Commentary by JOSEPH B. MAYOR, M.A., Professor of Classical Literature at King's College, London, formerly Fellow and Tutor of St John's College, Cambridge, together with a new collation of several of the English MSS. by J. H. SWAIN- SON, M.A., formerly Fellow of Trinity College, Cambridge. \Nearly Ready. MATHEMATICS, PHYSICAL SCIENCE, &c. THE ELECTRICAL RESEARCHES OF THE HONOURABLE HENRY CAVENDISH, F.R.S. Written between 1771 and 1781, Edited from the original manuscripts in the possession of the Duke of Devonshire, K. G., by J. Clerk Maxwell, F.R.S. Demy 8vo. cloth. i8j. "This work, which derives a melancholy endish's results with those of modem expcri- interest from the lamented death of the editor menters. In some instances they describe following so closely upon its publication, is a experiments undertaken by the editor for the valuable addition to the history of electrical express purpose of throwing light on Caven- research. . . . The papers themselves are most dish's methods of investigation. Every de- carefully reproduced, wth fac-similes of the partment of editorial duty appears to have author's sketches of experimental apparatus. been most conscientiously performed ; and it A series of notes by the editor are appended, must have been no small satisfaction to Prof, some of them devoted to mathematical dis- Maxwell to see this goodly volume completed cussions, and others to a comparison of Cav- beforehisUfe's work was done." — AiJieneEunt. A TREATISE ON NATURAL PHILOSOPHY. By Sir \V. Thomson, LL.D., D.C.L., F.R.S., Professor of Natural Philosophy in the University of Glasgow, and P. G. Tait, M.A., Professor of Natural Philosophy in the University of Edinburgh. Vol. I. Part I. 16s. " In this, the second edition, we notice a could form within the time at our disposal large amount of new matter, the importance would be utterly inadequate." — Nature, of which is such that any opinion which we ELEMENTS OF NATURAL PHILOSOPHY. By Professors Sir W. Thomson and P. G. Tait. Pait I. 8vo. cloth. Second Edition, gj. " This worlc is designed especially for the trigonometry. Tyros in Natural Philosophy use of schools and junior classes in the Uni- cannot be better directed than by being told versifies, the mathematical methods being to give their diligent attention to an intel- limited almost without exception to those of ligent digestion of the contents of this excel- the most elementary geometry, algebra, and lent vade »ucum."—IroK. London: Cambridge Warehouse, ij Paternoster Row. 12 PUBLICATIONS OF A TREATISE ON THE THEORY OF DETER- MINANTS AND THEIR APPLICATIONS IN ANALYSIS AND GEOMETRY, by Robert Forsyth Scott, M.A., of St John's College, Cambridge. [Nearly ready. HYDRODYNAMICS, A Treatise on the Mathematical Theory of the Motion of Fluids, by Horace Lamb, M.A., formerly Fellow of Trinity College, Cambridge; Professorof Mathematics in the University of Adelaide. DemySvo. I2j. THE ANALYTICAL THEORY OF HEAT, By Joseph Fourier. Translated, with Notes, by A. Freeman, M.A. Fellow of St John's College, Cambridge. Demy Octavo. i6j. **Fourier*s treatise is one of the very few matics who do not follow with freedom a scientific books which can never be rendered treatise in any language but their own. It antiquated by the progress of science. It is is a model of mathematical reasoning applied not only the first and the greatest book on to physical phenomena, and is remarkable for the physical subject of the conduction of the ingenuity of the analytical process cm- Heat, but in every Chapter new views are ployed by the author." — Contemporary opened up into vast fields of mathematical Remew, October, 1878. speculation, "There cannot be two opinions as to the " Whatever text-books may be written, value and importance of the Thiorie de la giving, perhaps, more succinct proofs of Chaleur. It has been called 'an exquisite Fourier's different equations, Fourier him- mathematical poem,' not once but many times, self will in all time coming retain his unique independently, by mathematicians of different prerogative of being the guide of his reader schools. Many of the very greatest of mo- into regions inaccessible to meaner men, how- dem mathematicians regard it, justly, as the ever expert." — Extract front letter oj" Pro- key which first opened to them the treasure- Jessor Clerk Maxwell. house of mathematical physics. It is still the "It is time that Fourier's masterpiece, text-book of Heat Conduction, and there The Analytical Theory of Heat, trans- seems little present prospect of its being lated by Mr Alex. Freeman, should be in- superseded, though it is already more than troduced to those English students of Mathe- half a century old." — Nature. MATHEMATICAL AND PHYSICAL PAPERS, By George Gabriel Stokes, M.A., D.C.L., LL.D., F.R.S., Fellow of Pembroke College, and Lucasian Professor of Mathematics in the University of Cambridge. Reprinted from the Original Journals and Transactions, withAdditionalNotesbytheAuthor. Vol. I. {Nearlyready. An elementary TREATISE on QUATERNIONS, By P. G. Tait, M.A., Professor of Natural Philosophy in the Univer- sity of Edinburgh. Second Edition. Demy 8vo. 14J. COUNTERPOINT. A Practical Course of Study, by Professor G. A. Macfarren, M.A., Mus. Doc. Second Edition, revised. Demy Quarto, cloth, "js. 6d. A CATALOGUE OF AUSTRALIAN FOSSILS (including Tasmania and the Island of Timor), Stratigraphically and Zoologically arranged, by ROBERT Etheridge, Jun., F.G.S., Acting Paleontologist, H.M. Geol. Survey of Scotland, (formerly Assistant- Geologist, Geol. Survey of Victoria). Demy Ofbavo, cloth, lar. 6d. "The work is arranged with great dear- papers consulted by the author, and an index ness, and contains a full list of.the books and to the genera." — Saturday Review. ILLUSTRATIONS OF COMPARATIVE ANA- TOMY, VERTEBRATE AND INVERTEBRATE, for the Use of Students in the Museum of Zoology and Comparative Anatomy. Second Edition. Demy Octavo, cloth, 2j. 6d. London: Cambridge Warehouse, 17 Paternoster Row. THE CAMBRIDGE UNIVERSITY PRESS. 13 A SYNOPSIS OF THE CLASSIFICATION OF THE BRITISH PALEOZOIC ROCKS, by the Rev. Adam Sedgwick, M.A., F.R.S., and Frederick M^COY, F.G.S. One vol., Royal Quarto, Plates, ^i. \s. A CATALOGUE OF THE COLLECTION OF CAMBRIAN AND SILURIAN FOSSILS contained in the Geological Museum of the University of Cambridge, by J. W. Salter, F.G.S. With a Portrait of Professor Sedgwick. Royal Quarto, cloth, 7J-. bd. CATALOGUE OF OSTEOLOGICAL SPECIMENS contained in the Anatomical Museum of the University of Cam- bridge. Demy Oftavo. is. 6d. THE MATHEMATICAL WORKS OF ISAAC BARROW, D.D. Edited by W. Whewell, D.D. Demy Octavo. Js. 6d. ASTRONOMICAL OBSERVATIONS made at the Observatory of Cambridge by the Rev. James Challis, M.A., F.R.S., F.R.A.S., Plumian Professor of Astronomy and Experi- mental Philosophy in the University of Cambridge, and Fellow of Trinity College. For various Years, from 1846 to i860. ASTRONOMICAL OBSERVATIONS from 1861 to 1865. Vol. XXI. Royal 4to. cloth. 15^. LAW. A SELECTION OF THE STATE TRIALS. By J. W. Willis-Bund, M.A., LL.B., Barrister-at-Law, Professor of Constitutional Law and History, University College, London. Vol. 1. Trials for Treason (1327 — 1660). Crown 8vo. cloth, i8j. " A great and good service has been done points in what may be called the growth of to all students of history, and especially to the Law of Treason which he wishes to those of them who look to it in a legal aspect, bring clearly under the notice of the student, by Prof. J. W. Willis- Bund in the publica- and the result is, that there is not a page in tion of a Selection of Cases from the State the book which has not its own lesson Trials. . . . Professor Willis- Bund has been In all respects, so far as we have been able very careful to give such selections from the to test it, this book is admirably done." — State Trials as will beat illustrate those Scotsman. THE FRAGMENTS OF THE PERPETUAL EDICT OF SALVIUS JULIANUS, collected, arranged, and annotated by Bryan Walker, M.A. LL.D., Law Lecturer of St John's College, and late Fellow of Corpus Christi College, Cambridge. Crown 8vo., Cloth, Price 6s. " This is one of the latest, wo believe mentaries and the Institutes . . . Hitherto Quite the latest of the contributions made to the Edict has been almost inaccessible to lesal scholarship by that revived study of the ordinary English student, and such a the Roman Law at Cambridge which is now student will be mterested as well as perhaps so marked a feature in the industrial life surprised to find how abundantly the extant of the University . . In the present book fragments illustrate and clear up pomts which we have the fruits of the same kind of have attracted his attention in the Commen- thorough and well-ordered study which was taries, or the Institutes, or the Digest. "- brought to bear upon the notes to the Com- Law Times. London: Cambridge Warehouse, 17 Paternoster Row. 14 PUBLICATIONS OF THE COMMENTARIES OF GAIUS AND RULES OF ULPIAN, (New Edition, revised and enlarged.) With a Translation and Notes, by J. T. Abdy, LL.D., Judge of County- Courts, late Regius Professor of Laws in the University of Cambridge and Bryan Walker, M.A., LL.D., Law Lecturer of St John's College, Cambridge, formerly Law Student of Trinity Hall and Chancellor's Medallist for Legal Studies. Crown 0(flavo, i6j. "As scholars and as editors Messrs Abdy "The number of books on various subjects and Walker have done their work well. of the civil law, which have lately issued from For one thing the editors deserve the Press, shews that the revival of the study special commendation. They have presented of Roman jurisprudence in this country is Gaius to theVeader with few notes and those genuine and increasing. The present edition merely by way of reference or necessary of Gaius and Ulpian from the Cambridge explanation. Thus the Roman jurist is University Press indicates that the Universi- allowed to speak for himself, and the reader ties are alive to the importance of the move- feels that he is really studying Roman law ment."— Z.aw younial, in the original, and not a fanciful representa- tion of it," — Athen^ufn. THE INSTITUTES OF JUSTINIAN, translated with Notes by J. T. Abdy, LL.D., Judge of County Courts, late Regius Professor of Laws in the University of Cambridge, and formerly Fellow of Trinity Hall ; and Bryan Walker, M.A., LL.D., Law Lecturer of St John's College, Cambridge ;. late Fellow and Lecturer of Corpus Christi College \ and formerly Law Student of Trinity Hall. Crown 06lavo, i6j'. "We welcome here a valuabl-e contribution attention is distracted from the subject-matter to the study of jurisprudence. The text of by the difficulty of struggling through the the /wj/zVzi^^j' is occasionally perplexing, even language in which it is contained, it will be to practised scholars, whose knowledge of almost indispensable." — Spectator. classical models does not always avail them " The notes are learned and carefully com- in dealing with the technicalities of legal piled, and this edition will be found useful phraseology. Nor can the ordinary diction- to students." — Law Times. aries be expected to furnish all the help that " Dr Abdy and Dr Walker have produced is wanted. This translation will then be of a book which is both elegant and useful." — great use. To the ordinary student, whose Athejixuin, SELECTED TITLES FROM THE DIGEST, annotated by B. Walker, M.A., LL.D. Part L Mandati vel Contra. Digest XVll. i. Crown 8vo., Cloth, is. "This small volume is published as an ex- say that Mr Walker deserves credit for the periment. The_ author proposes to publish an way in which he has performed the task un- annotated edition and translation of several dertaken. The translation, as might be ex- books of the Digest if this one is received pected, is sohoiarly." Law Times. with favour. We are pleased to be able to Part IL De Adquirendo rerum dominio and De Adquirenda vel amit- tenda possessione. Digest XLI. i & 1 1, {Nearly Ready. GROTIUS DE JURE BELLI ET PACIS, with the Notes of Barbeyrac and others ; accompanied by an abridged Translation of the Text, by W. Whewell, D.D. late Master of Trinity College. 3 Vols. Demy 06tavo, I2j. The translation separate, 6j. London: Cambridge Warehouse, 17 Paternoster Row. THE CAMBRIDGE UNIVERSITY PRESS, 15 HISTOEY. LIFE AND TIMES OF STEIN, OR GERMANY AND PRUSSIA IN THE NAPOLEONIC AGE, by J. R. Seeley, M.A.5 Regius Professor of Modern History in the University of Cambridge, with Portraits and Maps. 3 Vols. Demy 8vo. 48^-. " If we tould conceive anything similar to a protective system in the intellectual de- partment, we might perhaps look forward to a time when our historians would raise the cry of protection for native industry. Of the unquestionably greatest German men of modern history — I speak of Frederick the Great, Goethe and Stein — the first two found long since in Carlyle and Lewes biographers who have undoubtedly driven their German competitors out of the field. And now in the year just past Professor Seeley of Cambridge has presented us with a biography of Stein which, though it modestly declines competi- tion with German works and disowns the presumption of teaching us Germans our own history, yet casts into the shade by its bril- liant superiority all that we have ourselves hitherto written about Stein,... In five long chapters Seeley expounds the legislative and administrative reforms, the emancipation of the person and the soil, the beginnings of free administration and free trade, in short the foundation of modern Prussia, with more exhaustive thoroughness, with more pene- trating insight, than any one had done be- fore." — Deutsche Rundschau. " Dr Busch's volume has made people think and talk even more than usual of Prince Bismarck, and Professor Seeley's very learned work on Stein will turn attention to an earlier and an almost equally eminent German states- man It is soothing to the national self-respect to find a few Englishmen, such as the late Mr Lewes and Professor Seeley, doing for German as well as English readers what many German scholars have done for us." — Times. " In a notice of this kind scant justice can be done to a work like the one before us ; no short riswmi can give even the most meagre notion of the contents of these volumes, which contain no pa^e that is superfluous, and none that is unmteresting To under- stand the Germany of to-day one must study the Germany of many yesterdays, and now that study has been made easy by this work, to which no one can hesitate to assign a very high place among those recent histories which have aimed at original research." — Athe- fusujn. "The book before us fills an important gap in English — nay, European — historical literature, and bridges over the history of Prussia from the time of Frederick the Great to the days of Kaiser Wilhelm. It thus gives the reader standing ground whence he may regard contemporary events in Germany in their proper historic light We con- gratulate Cambridge and her Professor of History on the appearance of such a note- worthy production. And we may add that it is something upon which we may congratulate England that on the especial field of the Ger- mans, history, on the history of their own country, by the use of their own literary weapons, an Englishman has produced a his- tory of Germany in the Napoleonic age far superior to any that exists in German." — Ejcayniner. THE UNIVERSITY OF CAMBRIDGE FROM THE EARLIEST TIMES TO THE ROYAL INJUNCTIONS OF 1535, by James Bass Mullinger, M.A. Demy 8vo. cloth (734 pp.), 12s. "We trust Mr Mullinger will yet continue his history and bring it down to our own day. " — A cademy. *'He has brought together a mass of in- structive details respecting the rise and pro- gress, not only of his own University, but of all the principal Universities of the Middle Ages We hope some day that he may continue his labours, and give us a history of HISTORY OF THE COLLEGE OF ST JOHN THE EVANGELIST, by Thomas Baker, B.D., Ejected Fellow. Edited by John E. B. Mayor, M.A., Fellow of St John's. Two Vols. Demy 8vo. 24^. and academical , who have hitherto had to be content with 'Dyer.'" — Academy. It may be thought that the history of a the University during the troublous times of the Reformation and the Civil War." — Atht- n^um. "Mr MuUinger^s work is one of great learning and research, which can hardly fail to become a standard book of reference on the subject. . , . We can most strongly recom- mend this book to our readers." — Spectator. "To antiquaries the book will be a source of almost inexhaustible amusement, by his- torians it will be found a work of considerable service on questions respecting our social progress in past times ; and the care and thoroughness with which Mr Mayor has dis- charged his editorial functions are creditable to his learning and industry." — Atken^um. " The work displays very wide reading, and it will be of great use to members of the college and of the university, and, perhaps, of stil greater use to students of English history, ecclesiastical, political, social, literary college cannot beparticularlyattractive. The two volumes before us, however, have some- thing more than a mere special interest for those who have been in any way connected with St John's College, Cambridge; they contain much which will be read with pleasure by a far wider circle... The index with which Mr Mayor has furnished this useful work leaves nothing to be desired." — Spectator. London: Cambridge Warehouse, 17 Paternoster Row, i6 PUBLICATIONS OF HISTORY OF NEPAL, translated by MuNSHi Shew Shunker Singh and Pandit Shri GuNANAND ; edited with an Introductory Sketch of the Country and People by Dr D. Wright, late Residency Surgeon at Kathmandu, and with facsimiles of native drawings, and portraits of Sir Jung Bahadur, the King of Nepal, &c. Super-royal 8vo. Price lis. " The Cambridge University Press have done well in publishing this work. Such translations are valuable not only to the his- torian but also to the ethnologist; Dr Wright's Introduction is based on personal inquiry and observation, is written intelli- gently and candidly, and adds much to the Tralue of the volume. The coloured litho- graphic plates are interesting." — Nature. ** The history has appeared at a very op- portune moment.. .The volume,, .is beautifully printed, and supplied with portraits of Sir Jung Bahadoor and others, and with excel- lent coloured sketches illustrating Nepaulese architecture and religion.*' — Examiner. " In pleasing contrast with the native his- tory are the five introductory chapters con- tributed by Dr Wright himself, who saw as much of Nepal during his ten years' sojourn as the strict rules enforced against foreigners even by Jung Bahadur would let him see." — Indian Mail. *'Von nicht geringem Werthe dagegen sind die Beigaben, welche Wright als 'Appendix' hinter der 'history' folgen lasst, Aufzah- lungen namlich der in Nepal iiblichen Musik- Instrumente, Ackergerathe, Miinzen, Ge- wichte, Zeittheilung, sodann ein kurzes Vocabular in Parbatiya und Newslri, einige Newari songs mit Interlinear-Uebersetzung, eine Konigsliste, und, last not least, ein Verzeichniss der von ihm mitgebrachten Sanskrit-Mss., welche jetzt in der Universi- tats-Bibliothek in Cambridge deponirt sind,** — A. Weber, Liter aturzeitungy Jahrgang 1877, Nr. 26. "On trouve le portrait et la g^n^alogie de Sir Jang Bahadur dans I'excellent ouvrage que vient de publier Mr Daniel Wright, sous le titre de * History of Nepal, translated from the Parbatiya, etc.'" — M. Garcin de Tassv in La Langueet la Littirature Hin- doustanies in 1877. Paris, 1878. SCHOLAR ACADEMICAE: Some Account of the Studies at the English Universities in the Eighteenth Century. By Christopher Wordsworth, M.A., Fellow of Peterhouse ; Author of " Social Life at the Enghsh Universities in the Eighteenth Century." Demy octavo, cloth, 15^-, teresting, and instructive. Among the mat- ters touched upon are Libraries, Lectures, the Tripos, the Trivium, the Senate House, the Schools, text-books, subjects of study, foreign opinions, interior life. We learn even of the various University periodicals that have had their day. And last, but not least, we are given in an appendix a highly interesting series of private letters from a Cambridge student to John Strype, giving a vivid idea of life as an undergraduate and afterwards, as the writer became a graduate and a fellow." — University Magazine. "Only those who have engaged in like la- bours will be able fully to appreciate the sustained industry and conscientious accuracy discernible in every page. . . . Of the whole volume it may be said that it is a genuine service rendered to the study of University history, and that the habits of thought of any writer educated at either seat of learning in the last century will, in many cases, be far better understood after a consideration of the materials here collected." — Academy. "The general object of Mr Wordsworth's book is sufficiently apparent from its title. He has collected a great quantity of minute and curious information about the working of Cambridge institutions in the last century, with an occasional comparison of the corre- sponding state of things at Oxford. It is of course impossible that a book of this kind should be altogether entertaining as litera- ture. To a great extent it is purely a book of reference, and as such it will be of per- manent value for the historical knowledge of English education and learning." — Saturday Review. "In the work before us, which is strictly what it professes to be, an account of university stu- dies, we obtain authentic information upon the course and changes of philosophical thought in this country, upon the general estimation of letters, upon the relations of doctrine and science, upon the range and thoroughness of education, and we may add, upon the cat- like tenacity of life of ancient forms,... The particulars Mr Wordsworth gives us in his excellent arrangement are most varied, in- THE ARCHITECTURAL HISTORY OF THE UNIVERSITY AND COLLEGES OF CAMBRIDGE, By the late Professor Willis, M.A. With numerous Maps, Plans and Illustrations. Continued to the present time, and edited by John Willis Clark, M.A., formerly Fellow of Trinity College, Cambridge. [/« the Press. London: Cambridge Warehouse, 17 Paternoster Row. THE CAMBRIDGE UNIVERSITY PRESS. 17 MISCELLANEOUS. STATUTA ACADEMIC CANTABRIGIENSIS. Demy Odlavo. 2j. sewed, ORDINATIONES ACADEMIC CANTABRIGIENSIS Demy Oflavo, cloth, y. 6d. TRUSTS, STATUTES AND DIRECTIONS affecting (i) The Professorships of the University. (2) The Scholarships and Prizes. (3) Other Gifts and Endowments. Demy 8vo. 5J. COMPENDIUM OF UNIVERSITY REGULATIONS, for the use of persons in Statu Pupillari. Demy Odlavo. 6d. CATALOGUE OF THE HEBREW MANUSCRIPTS preserved in the University Library, Cambridge. By Dr S. M. SCHILLER-SZINESSY. 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Perowne, D.D., Dean of Peterborough. The want of an Annotated Edition of the Bible, in handy portions, suitable for School use, has long been felt. In order to provide Text-books for School and Examination pur- poses, the Cambridge University Press has arranged to publish the several books of the Bible in separate portions at a moderate price, with introductions and explanatory notes. The Very Reverend J. J. S. Perowne, D.D., Dean of Peter- borough, has undertaken the general editorial supervision of the work, and will be assisted by a staff of eminent coadjutors. Some of the books have already been undertaken by the following gentlemen : Rev. A. Carr, M.A., Assistant Master at Wellington College. Rev. T. K. Cheyne, Fellow of Balliol College, Oxford. Rev. S. Cox, Nottingham. Rev. A. B. Davidson, D.D., Professor of Hebrew, Edinburgh. Rev. F. W. Farrar, D.D., Canon of Westminster. Rev. A. E. Humphreys, M.A., Fellow of Trinity College, Cambridge, Rev. A. F. 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Whitaker, M.A., Fellow of St John's College, Cambridge. Now Ready. Cloth, Extra Fcap. 8vo. THE BOOK OF JOSHUA. Edited by Rev. G. F. Maclear, D.D. With 2 Maps. -is. 6d. THE BOOK OF JONAH. By Archdn. Perowne. is. bd. London: Cambridge Warehouse, 17, Paternoster Row. THE CAMBRIDGE UNIVERSITY PRESS. 19 THE CAMBRIDGE BIBLE FOR SCHOOLS.— C(7«/2««^V given a good deal of historical informa- tion, in which they have, we think, done well. At the beginning of the book will be found excellent and succinct accounts of the constitution of the French army and Parliament at the period treated of." — Saturday Rci'tcv. "We are pleased to find, since we may not have the Cambridge work done by Cambridge men that the University has been so fortunate as to secure the sernces of ^Ir Gustave Masson, whose work is always admirable and thorough. On this occasion the Syndicate has allowed him to take an English cctlahor.il^ur in the person of Mr Prothero, examiner for the Historical Tripos at Cambridge. 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