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 The mathematical theory of the toPj-e!='" 
 
 3 1924 001 070 899 
 
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PRINCETON LECTURES. 
 
 A series of volumes containing the notable lectures de 
 
 livered c 't the occasion of the Sesquicentetiuial 
 
 celebration of Princeton University. 
 
 Tbe French Revolution and English Literature. Six Lectures. 
 
 By Prof, Edward Dowden, Trinity College, Dublin. 
 Theism. Two Lectures. By Prof. Andrew Seth, University of 
 
 Edinburgh. 
 The Discharge of Electricity In Gases. Four Lectures. By Prof. 
 
 J. J. Thomson, University of Cambridge, 
 The Mathematical Theory of the Top. Four Lectures. By Prof 
 
 Felix Klein, University of Gbttingen. 
 The Descent of the Primates. By Prof. A. A. W. Hubrecht, 
 
 University ol Utrecht. 
 
 The nature and Origin of the Koun Genders in the Indoger- 
 manic Languages. By Prof. Karl Brugmann, University of 
 Leipsic 
 
 The Claims of the Old Testament. Two Lectures By Prof. 
 Stanley Leathes, D.D., King's College, London. 
 
THE MATHEMATICAL THEORY 
 OF THE TOP 
 
THE 
 
 MATHEMATICAL THEORY 
 OF THE TOP 
 
 LECTURES DELIVERED ON THE OCCASION OF 
 
 THE SESQUICENTENNIAL CELEBRATION 
 
 OF PRINCETON UNIVERSITY 
 
 BT 
 
 FELIX KLEIN 
 
 Professor of Mathematics in the Univeesttt of Gottikgen 
 
 WITH ILLUSTRATIONS 
 
 NEW YORK 
 
 CHARLES SCRIBNER'S SONS 
 
 1897 
 
 EH 
 
Copyright, 1897, 
 By Charles Scribnek's Sons. 
 
 WoriHooB 3|Hfsa 
 
 J. S. Cuihing & Co. — Berwick fc Smith 
 Norwood Mass. U.S.A. 
 
NOTE 
 
 These lectures on the analytical formulae relat- 
 ing to the motion of the top were delivered on 
 Monday, Tuesday, Wednesday, and Thursday, 
 October 12-15, ^896. They were reported and 
 prepared in manuscript form by Professor H. B. 
 Pine of Princeton University, and the manuscript 
 was revised by Professor Klein. 
 
LECTURE I 
 
 In the following lectures it is proposed to con- 
 sider certain interesting and important questions 
 of dynamics from the standpoint of the theory of 
 functions of the complex variable. I am to de- 
 velop a new method, which, as I think, renders 
 the discussion of these questions simpler and 
 more attractive. My object in presenting it, how- 
 ever, is more general than that of throwing light 
 on a particular class of problems in dynamics. 
 I wish by an illustration which may fairly be 
 regarded as representative to make evident the 
 advantage which is to be gained by dynamics 
 and astronomical and physical science in general 
 from a more intimate association with the modern 
 pure mathematics, the theory of functions espe- 
 cially. 
 
 I venture to hope, therefore, that my lectures 
 may interest engineers, physicists, and astronomers 
 as well as mathematicians. If one may accuse 
 mathematicians as a class of ignoring the mathe- 
 matical problems of the modern physics and astron- 
 omy, one may, with no less justice perhaps, accuse 
 
Z MOTION OP THE TOP 
 
 physicists and astronomers of ignoring departments 
 of the pure mathematics which have reached a 
 high degree of development and are fitted to ren- 
 der valuable service to physics and astronomy. It 
 is the great need of the present in mathematical 
 science that the pure science and those depart- 
 ments of physical science in which it finds its most 
 important applications should again be brought 
 into the intimate association which proved so fruit- 
 ful in the work of Lagrange and Gauss. 
 
 I shall confine my discussion mainly to the 
 problem presented in the motion of a top — 
 meaning for the present by "top" a rigid body 
 rotating about an axis, when a single point in 
 this axis, not the centre of gravity, is fixed in 
 position. 
 
 In the present lecture I shall present some 
 preliminary considerations of a purely geometri- 
 cal character. But it is necessary first of all to 
 obtain an analytical representation of the rotation 
 of a rigid body about a fixed point, and I shall 
 begin with a statement of the methods ordinarily 
 used. 
 
 We introduce two systems of rectangular axes 
 both having their origin at the fixed point: the 
 one system, x, y, z, fixed in space ; the other, X, 
 Y, Z, fixed in the rotating body. Then the ordi- 
 nary equations of transformation from the one 
 
LECTURE I 
 
 system to the other, which may be exhibited in 
 the scheme : 
 
 X Y Z 
 
 X 
 
 a 
 
 b 
 
 c 
 
 a' 
 
 b' 
 
 c' 
 
 a" 
 
 b" 
 
 c" 
 
 (1) 
 
 give at once, when the nine direction cosines, 
 a, b, c, a', ■•• are known functions of the tinie t, 
 the representation of the motion of the movable 
 system X, Y, Z, with respect to the fixed system 
 X, y, z. 
 
 As is well known, these cosines are not inde- 
 pendent; they are rather functions of but three 
 independent quantities or parameters. It is cus- 
 tomary to employ one or other of the following 
 sets of parameters, both of which were introduced 
 by Euler. 
 
 The first set of parameters, which is non-sym- 
 metrical, consists of the angle i* which the iJ-axis 
 makes with the »-axis, and the angles </> and \fi, 
 which the line of intersection of the xy- and XY- 
 planes makes with the X-axis and the a^axis 
 respectively. Because of the frequent use made 
 of these parameters in astronomy, I shall call 
 them the "astronomical parameters." When the 
 cosines a, b, c, ■•• have been expressed in terms 
 
4 MOTION OF THE TOP 
 
 of them, the orthogonal substitution (1) be- 
 comes : 
 
 X Y Z 
 
 X 
 
 (2) y 
 
 cos COS i(/ — COS i> sin ^ sin ifi, — sin ^ cos t^ — cos iV cos sin i/(, sin i? sin i(i 
 cos 1^ sin i)/ + cos iV sin ^ cos i/(, — sin (^ sin i(( + cos It cos i^ cos ^, — sin ,J cos <| 
 sin//sin0 , sin,5icoS(^ , cosiJ 
 
 The second set of parameters may be defined 
 as follows. Every displacement of our body is 
 equivalent to a simple rotation about a fixed axis. 
 Let to be the angle of rotation, and a, h, c the 
 angles which the axis makes with OX, OY, OZ; 
 and set 
 
 A = cos a sin -, B = cos 6 sin -, O = cos c sin -, D = cos -. 
 2 2 2 2 
 
 The quantities A, B, C, D (of which but three 
 
 are independent, since, as will be seen at once, 
 
 J^ + B^+C-^-D'^ 1) are the parameters under 
 
 consideration. In terms of them our orthogonal 
 
 substitution (1) is 
 
 X Y Z 
 
 X 
 
 (3) y 
 
 2{AB+CD), D^ + B'^0^-A^,2(BC-AD) 
 2(iAC-BD), 2(BC+An) D'^+C^-A^-B' 
 
 or, if use be not made of the relation 
 A^ + B' + C^ + D'' = 1, 
 
 a substitution with these coefficients each divided 
 by A' + B'+C' + D\ I shall call these the 
 
LECTURE I 5 
 
 "quaternion parameters," inasmuch as the qua- 
 ternionists make frequent use of them. The 
 quaternion corresponding to our rotation is 
 
 q = D + iA+jB-\-kC. 
 
 These parameters are very symmetrical, and for 
 that reason very attractive. Nevertheless, they 
 do not prove to be the most advantageous system 
 for our present purpose. Our problem is not a 
 symmetrical problem. In it one of the axes, Oz, 
 in the direction of gravity, plays an exceptional 
 role ; the motion of the top is not isotropic. 
 
 Instead of either of these commonly used sys- 
 tems of parameters, I propose to introduce another, 
 which so far as I know has not yet been employed 
 in dynamics. 
 
 Let X, y, z be the coordinates of a point on a 
 sphere fixed in space which has the radius r and 
 the centre 0, and X, Y, Z the coordinates of a 
 point on a sphere congruent with the first but fixed 
 in the rotating body. As the body rotates, the 
 second sphere slides about on the first, but remains 
 always in congruence with it. 
 
 It is characteristic of every point on the first 
 sphere that the relation 
 
 x + iy _ r + z 
 r — z x—iy 
 
 holds good between its coordinates. 
 
b MOTION OF THE TOP 
 
 If we represent the values of these equal ratios by 
 ^, obviously ^ is a parameter for the points of the 
 sphere, which completely determines one of these 
 points for every value that it may take. Thus the 
 upper extremity of the «-axis is characterized by 
 the value oo of ^, the lower extremity by the value 
 ; to real values of ^ correspond the points on the 
 great circle of the sphere in the plane y = 0, and to 
 pure imaginary values the points of the great circle 
 in the plane a; = 0. 
 
 For the points of the second sphere, in like man- 
 ner, there is a parameter Z connected with the co- 
 ordinates X, Y, Z by the equations, 
 
 X+iT ^ r+Z ^ 
 r-Z X-iY ' 
 
 which defines these points as t, defined the points 
 of the fixed sphere. 
 
 If now i, and Z be parameters of corresponding 
 points on the two spheres, what is the relation be- 
 tween these parameters when the second sphere is 
 subjected to a rotation ? It is a simple linear rela- 
 tion of the form 
 
 ^ yZ-)-8' 
 
 in which a, ji, y, 8 are themselves in general complex 
 quantities, but so related that a is the conjugate tm- 
 
LECTURE I 7 
 
 aginary to S, and /3 to — y ; or, adopting the ordi- 
 nary notation, a = 8 and ^ = — y. 
 
 It is obvious, a priori, that the relation must be 
 linear, and a very simple reckoning such as I have 
 given in my treatise on the Icosahedron (p. 32) 
 establishes the special relations among the coefB.- 
 cients. There are but four real quantities in- 
 volved in a, j3, y, 8, only the ratios of which 
 need be considered independent, since these ratios 
 alone appear in the expression for ^ ; unless, as is 
 generally more convenient, we introduce the fur- 
 ther relation aB — fiy — 1. 
 
 It is these quantities a, /3, y, 8 connected by the 
 relation a8 — /3y = 1, which together with ^ we pro- 
 jjose to use as our parameters in the discussion of 
 the problem now under consideration. They were 
 introduced into mathematics by Eiemann forty 
 years ago, and have proved to be peculiarly useful 
 in different geometrical problems intimately con- 
 nected with the theory of functions, especially in 
 the theory of minimal surfaces and the theory of 
 the regular solids. We hope to show that they may 
 be employed to quite as great advantage in the study 
 of all problems connected with the motion of a rigid 
 body about a fixed point. 
 
 Corresponding to the orthogonal substitution (1), 
 we have in terms of our new parameters the substi- 
 tution 
 
«= 
 
 2al3 
 
 1^ 
 
 ay 
 
 a8 + ^y 
 
 ps 
 
 y' 
 
 2yS 
 
 S' 
 
 MOTION OF THE TOP 
 
 X+iT -Z -X+iT 
 
 x + iy 
 (4) 
 
 ■x + iy 
 
 as may be demonstrated without serious reckoning 
 as follows. And I may remark incidentally that it 
 seems to me better wherever possible to effect a 
 mathematical demonstration by general considera- 
 tions which bring to light its inner meaning rather 
 than by a detailed reckoning, every step in which 
 the mind may be forced to accept as incontro- 
 vertible, and yet have no understanding of its real 
 significance. 
 
 Consider the sphere of radius 0, 
 
 x' + y^ + z'' = 0. 
 
 It is an imaginary cone whose generating lines 
 join the origin to the so-called "imaginary circle 
 at infinity,'' the circle in which all spheres inter- 
 sect at infinity. Por this sphere, 
 
 ,_ x + iy _ 2_ 
 
 — 2 x — iy 
 
 or X -\- iy : — z : X — iy =^^ : ^•. —1. 
 
 Here to each value of the parameter ^ there cor- 
 responds a single (imaginary) generating line of 
 the cone, and vice versa. In other words, there is 
 a relation of one-to-one correspondence between 
 
LBCTUEB I 9 
 
 the (imaginary) generating lines of the cone and 
 the values of ^, or the cone is unicursal. 
 
 There is, of course, the same relation between 
 the generating lines of the congruent cone 
 
 X^+Y^+Z' = 0, 
 
 which is fixed in the moving body, and the pa- 
 
 rameter ^^X+iY_ Z 
 
 X-iY 
 
 When the body rotates, this cone is simply car- 
 ried over into itself, so that the generating lines 
 in their new position are in one-to-one corre- 
 spondence with the same generating lines in their 
 original position. Between the parameters Z and 
 ^, which correspond to the generating lines in these 
 two positions, there is, therefore, also a relation of 
 one-to-one correspondence, or the two are connected 
 linearly, i.e. by a relation of the form : 
 
 ^ _ ceZ -I- j8 
 yZ + S' 
 
 where, as above, we suppose 
 
 aS-Py = 1. 
 
 If now we avail ourselves of the advantages to 
 be had from the use of homogeneous equations 
 and substitutions by replacing 
 
 C by {5, and Z by |, 
 
10 MOTION OF THE TOP 
 
 this single equation may be replaced by the two 
 homogeneous equations : 
 
 L = yZi + SZ2, 
 
 and the equations connecting x, y, z, and f, and 
 X, T, Z, and Z become, 
 
 a; + I?/ : - 2 : - a; + iV = ^1" : CiCa : f 2^ 
 X+ iY: -Z: -X+iY= Z^ : Z^Z^ : 1l 
 
 From these equations it follows that 
 
 a; + iy = a\X ■\-iY)+2 aj8(- Z)^ ^{- X+ iY) 
 
 -» = «y(X+iF) + (a8 + j8y)(-^) + |88(-X+iF) 
 
 -a; + !2/ = /(X + ir) + 2y8(-Z) + 8X-X+JF). 
 
 For it is immediately obvious that a; + iy is pro- 
 portional to ^1", therefore to 
 
 a^Z^ + 2 «;8ZiZ2 + /?%=, 
 
 and therefore finally to 
 
 and in like manner, that — z and —x + iy are pro- 
 portional to 
 
 ay (X+iY) + (a8 + /Jy)(- Z)-f- i88(- X+ iT), 
 
 and 
 
 Y'{X+iY) + 2yS(-Z)+B!'(-X+iY) 
 
LECTURE I .11 
 
 respectively. And that x + iy, — z, — a; 4- iy are 
 severally equal to these expressions and not merely 
 proportional to them, follows from the fact that the 
 determinant of the orthogonal substitution connect- 
 ing X, y, z with X, Y, Z must equal 1. 
 
 The demonstration, to be sure, applies directly 
 to the points of the imaginary cone only. But 
 it is known in advance that the transformation 
 which we are considering is a linear one for all 
 points of space. Its coefficients are the same for 
 all points, and we have merely availed ourselves 
 of the fact that the imaginary cone remains un- 
 changed by the transformation to determine them. 
 The same result might have been reached, though 
 
 less simply, by using the general formula ^ = ^ - 
 
 r — z 
 
 The equations (4), therefore, are those which con- 
 nect the coordinates of the initial and final posi- 
 tions of any point rigidly attached to the rotating 
 body. 
 
 The relations between our new parameters, a, 
 |8, y, 8, and the astronomical parameters, i?, 4>> "Aj 
 on the one hand, and the quaternion parameters 
 A, B, C, D, on the other, are of immediate in- 
 terest and of importance in the subsequent dis- 
 cussion. They are to be had very simply by a 
 comparison of the coefficients in the three schemes 
 (2), (3), (4), and, after reduction, prove to be ; 
 
12 MOTION OF THE TOP 
 
 (5) 
 
 and 
 
 a = COS ^ • e 2 , fi = ism'i-e ' , 
 
 ' a = D + iG, ji = -B + iA, 
 
 y = B + iA, S = D-iC. 
 
 Our new parameters are thus imaginary combina^ 
 tions of the real parameters in ordinary use. Mathe- 
 matical physics affords many examples of the ad- 
 vantage to be gained by employing such imaginary 
 combinations of real quantities. It is only necessary 
 to cite the use made of them in optics by Cauchy. 
 
 I may remark that Darboux in his Lemons sur 
 la tMorie ginirale des surfaces, Livre I., treats the 
 subject of rotation in a manner which is very simi- 
 lar to that which we have followed. But with him 
 the f itself is considered directly as a function of 
 the time and not the separate coefRcients a, p, y, 8. 
 His method thus lacks the simplicity which is pos- 
 sible when these are made the primary functions. 
 
 We now turn to a brief consideration of the 
 meaning of the substitution 
 
 yZ -f- 8' 
 
 when a, /S, y, 8 are still regarded as functions of 
 the time, but are general complex quantities, not 
 connected by the special relations a = 8, ^ = — y. 
 
LECTUKE I 
 
 13 
 
 We shall consider t also as capable of complex 
 values, not for the sake of studying the behavior 
 of a fictitious, imaginary time, but because it is 
 only by taking this step that it becomes possible 
 to bring about the intimate association of kinetics 
 and the theory of functions of a complex variable 
 at which we are aiming. 
 
 What is the meaning of the above formula? 
 It is still a real transformation of the sphere 
 on which we have defined i, into itself, a linear 
 transformation in which the coefficients are' all 
 real. 
 
 If the radius of the sphere be 1, as we shall 
 assume throughout the discussion of this general 
 transformation, or its equation when written homo- 
 geneously, be : 
 
 si? + y^ + z^ — f = 0, 
 
 the equations connecting x, y, z, t and X, Y, Z, 
 T are those indicated in the following scheme: 
 
 X+iY X-iY T+Z T-Z 
 
 x + iy 
 (6) X - iy 
 t + z 
 t — z 
 
 a8 
 
 ^y 
 
 ay 
 
 pi 
 
 yP 
 
 8a 
 
 ytt 
 
 ¥ 
 
 ap 
 
 Pa 
 
 dot 
 
 
 
 yS 
 
 Sy 
 
 yy 
 
 S8 
 
14 MOTION OF THE TOP 
 
 and when these equations are solved for x, y, z, t, 
 in terms of X, Y, Z, T, it will be found that the 
 coefiicients are real, as has been already stated. 
 
 This scheme may be derived in a manner analo- 
 gous to that followed in deriving the scheme (4). 
 
 The equation of the sphere 
 
 a? + y'' + z''-f=0, 
 or (x + iy)(x - iy) + (z + t)(z -t) = 0, 
 
 may, as is readily verified, be written in the form, 
 x+iy:x-iy:t + z:t-z = U2' ■ Ui ■ Ui ■ &C2', 
 where y = i, and ^Z, ^2' are, for real values of x, y, 
 z, t, the conjugate imaginaries to ^1, ^2 respectively. 
 
 As above, ^ = ^+M = l+±. 
 t — z X — iy 
 
 If then Zi, Z2, Z,', Z^ be quantities similarly 
 defined with respect to the movable sphere 
 
 we have corresponding to the transformation 
 
 . ^ ttZ + /3 ^ 
 7Z + 8 ' 
 
 the two pairs of equations : 
 
 ^r=aZ, + l3Z„ Cj' = aZ/ + ^Za', 
 
 ^2=7Z,+SZ2, f2' = yZi'+8Z2', 
 
 if the transformation is to be real. 
 
LfiCTUKE 1 15 
 
 And from this series of equations it follows by 
 the reasoning used on page 10 that x + iy is 
 equal to 
 
 al{X+iY)+!iy{X+iY)+ay{T + Z)+pl(T-Z), 
 
 and X — iy, t + z, t — z to the corresponding expres- 
 sions indicated in scheme (6). 
 
 The scheme (6) at once reduces to the scheme 
 (4) when the special supposition is made that 
 « = S and /3 = — y. And since this is the sufficient 
 and necessary condition that (6) reduce to (4), we 
 have here an independent demonstration that these 
 relations hold good among the parameters a, fi, y, S 
 when the motion is a rotation about a fixed point. 
 
 The general transformation (6) represents the 
 totality of those projective transformations or col- 
 lineations of space for which each system of gen- 
 erating lines of the sphere, x' -{-y^ + z^ — f = 0, is 
 transformed into itself, and among which all rota- 
 tions of the sphere are obviously included as 
 special cases. This is the geometrical meaning of 
 the equation 
 
 . _ «Z -f- ;8 
 
 yZ+a 
 
 for unrestricted values of a, j8, y, 8. 
 
 But the transformation admits also of a very 
 interesting kinematical interpretation which I shall 
 consider at length in my third lecture. With 
 
16 MOTION OP THE TOP 
 
 respect to it our sphere of radius 1 plays the r61e 
 of the fundamental surface or " absolute " in the 
 Cayleyan or hyperbolic non-Euclidian geometry. 
 For any free motion in such a space the absolute 
 remains fixed in position as in ordinary space the 
 imaginary circle at infinity x' + 'jf + z^ = 0, t = Q, 
 does, which is its absolute. 
 
 The transformation therefore represents a real free 
 motion in non^Eudidean space, and the six inde- 
 pendent real parameters involved in the ratios 
 a : j3 : y : 8 correspond to the oo° such possible 
 motions. Interpreted in Euclidean space, the trans- 
 formation represents a motion of the body com- 
 bined vifith a strain. 
 
 I close the present lecture with two remarks. 
 
 First, there is nothing essentially new in the con- 
 siderations with which we have been occupied thus 
 far. I have merely attempted to throw a method 
 already well known into the most convenient form 
 for application to mechanics. 
 
 Second, the non-Euclidean geometry has no metor 
 physical significance here or in the subsequent discus- 
 sion. It is used solely because it is a convenient 
 method of grouping in geometric form relations 
 which must otherwise remain hidden in formulas. 
 
LECTUEE II 
 
 I NOW proceed at once to the discussion of the 
 Lagrange equations of motion for our top, only- 
 pausing to remark once more that this problem of 
 the top is for us typical of all dynamical ques- 
 tions which are related to a sphere. To this cate- 
 gory belong also the problem of the spherical pendu- 
 lum (which in fact is a special case of the problem 
 of the top), the problem of the catenary on the 
 sphere, and all problems of the motion of a rigid 
 body about a fixed point. The simplest problem 
 of the type is that of the motion of a rigid body 
 about its centre of gravity, the Poinsot motion, as 
 we shall name it after Poinsot who treated it very 
 elegantly. 
 
 We shall first state the equations in terms of the 
 astronomical parameters; and to give the expres- 
 sions as simple a form as possible, I shall suppose 
 the principal moments of inertia of the top about 
 the fixed point of support each equal to 1. One 
 may call such a top a spherical top, as its momental 
 ellipsoid is a sphere. I wish it understood, how- 
 ever, that this restriction is not essential to the 
 c 17 
 
18 MOTIOX OF THE TOP 
 
 application of ovir method, but is rather made 
 solely for the sake of rendering its presentation 
 more easy. 
 
 On this assumption, we have for the kinetic 
 energy, T, of the motion the expression 
 
 r = i (<^'2 + .//'^ + 2 </> V cos * + ,y2)^ 
 
 where *', <^', i/r' are the derivatives of &, <^, ^ with 
 respect to t; and for the potential energy, V, the 
 expression 
 
 F=P cos .9, 
 
 where P represents the static moment of the top 
 with respect to 0. 
 
 The Lagrange equations are : 
 
 d^T_ ^hT ^ST 
 
 J*' = o Jf.=a 8'^'_ 8(r-F) 
 
 dt ' dt ' dt 8* 
 
 The first two equations are especially simple in 
 having their right members equal to zero, and 
 we are therefore able to derive immediately the 
 two algebraic first integrals 
 
 4>' + if/' cos i> = n, 
 f + (/)' cos & = l. 
 
 The quantities n and I are constants of integra- 
 tion, to be determined from the initial conditions 
 of the motion. In the following discussion we 
 shall suppose them positive. 
 
LEOTUKB II 19 
 
 In addition to these integrals, we have the equa- 
 tion of energy 
 
 T+V=li, 
 
 where h also is a constant determined, like I and n, 
 by the special conditions of the problem. 
 
 Solving the first two equations for <^' and xj/', and 
 substituting the results in the third, and setting 
 cos I? = M, and 
 
 U=2PvJ' - 2 7m2 + 2(ln-P)u + 21i-V- n\ 
 
 we obtain finally for t, <^, and i//, expressed as func- 
 tions of M, the formulas 
 
 . _ r du ,_ rn — lu du ._ ri—nu du 
 
 The problem of the motion of the top is thus re- 
 duced to three simple integrations or quadratures, 
 as indeed was demonstrated by Lagrange himself. 
 These integrals are elliptic integrals, U being 
 a polynomial of the third degree in u, the first an 
 elliptic integral of the "first kind" (which is 
 characterized by being finite for all values of the 
 independent variable), the remaining two elliptic 
 integrals of a more complex character. 
 
 It is often said that dynamics reached its ulti- 
 mate form in the hands of Lagrange, and the cry 
 "return to Lagrange" is frequently raised by 
 those who set little store by the value for physical 
 
20 MOTION OF THE TOP 
 
 science of recent developments in the pure mathe- 
 matics. But this is by no means just. Lagrange 
 reduced our problem to quadratures, but Jacobi 
 made a great stride beyo d him, as we mathema- 
 ticians think, by introducing the elliptic functions, 
 which enabled him to assign to t the r51e of inde- 
 pendent variable and to discuss the remaining varia- 
 bles M, </), i/r directly as functions of the time. An 
 advantage was thus gained not only for the under- 
 standing of the essential relations of the variables to 
 one another, hut for simplicity of computation also. 
 The coefficients a, b, c, a', •••, are uniform (or one 
 valued) functions of t, and one of the most useful 
 properties a function can possess, if its values must 
 be computed, is that it be uniform. This work of 
 Jacobi is not as well known as it should be, hav- 
 ing first appeared posthumously, in the second vol- 
 ume of his collected works, published by the Berlin 
 Academy in 1882. I may add that his pupils, Lott- 
 ner and Somoff, developed the same method in pa- 
 pers published in 1855 independently. It is shown 
 in these papers that the nine cosines a, b, c, •••, may 
 be expressed in terms of theta functions.* 
 
 * As is well known, Jacobi gave analogous formulas for 
 tlie nine cosines of the Poinsot motion in 1849. Closely re- 
 lated to this representation of the cosines is the interesting 
 theorem to which we shall return later on, that the motion 
 of our top may be reproduced by compounding two Poinsot 
 motions. 
 
LECTURE II 21 
 
 But while the a, b, c, ■••, considered as functions 
 of t, are much simpler than the integrals of La- 
 grange, they are at the same time much more com- 
 plicated than our parameters «, {3, y, S. These 
 parameters prove to be the simplest possible elliptic 
 functions of t; so that by introducing them we carry 
 to its completion the work begun by Jacobi, of re- 
 ducing our problem to its simplest elements. 
 
 For the proper understanding of this treatment 
 of the motion of the top, some knowledge of the 
 nature of elliptic functions is obviously necessary ; 
 and I know of no readier means of gaining this than 
 Riemann's metJiod of conformal representation, — of 
 which, moreover, we shall have other important 
 applications to make later on. 
 
 In accordance with this method, we construct 
 the " Eiemann surface " of the function y/U on the 
 plane of the complex variable u, in the follow- 
 ing manner : The polynomial U vanishes for three 
 values of u, all of which may readily be shown 
 to be real, and becomes infinite when m = oo. Two 
 of these roots, e,, e^, lie between — 1 and -1- 1 ; and 
 the third, e^, between -)- 1 and oo. Therefore, y/U 
 is a two-valued function of u everywhere in the 
 M-plane, except at the four points of the real axis, 
 ei, e^, e-i, e„. To obtain a surface, therefore, between 
 whose points and the values of Vf7 there shall be 
 a one-to-one correspondence, we lay over the it-plane 
 
22 
 
 MOTION OF THE TOP 
 
 two sheets, which are everywhere distinct except at 
 the points ej, e.^, e^, e„, in which they coalesce, and 
 associate with the points in the two sheets which 
 lie immediately over any point u in the M-plane, 
 the two corresponding values of VC/", one with each. 
 It will be found that if the point u describe any 
 simple circuit in the t(-plane, which encloses one 
 and but one of the points fij, 62, e^, e„, returning 
 finally to its initial position, V(7 will pass from 
 the one to the other of the two values which 
 correspond to the initial value of u; the point 
 corresponding to u in the Riemann surface of Vf7, 
 must, therefore, move from a position in the one 
 sheet to a position immediately under (or over) this 
 in the other. But this is possible only if we sup- 
 pose the two sheets to cross along some line running 
 out from each of the points, e^, 62, 63, e„, — not to 
 intersect, but to cross, as non-intersecting lines 
 in space may be said to cross. Inasmuch as this 
 is the simplest hypothesis possible, we shall take- 
 as these lines of crossing, in the present case, the 
 segments, 6162, Cge^, of the real axis ; and have, as 
 a rough representation of the Riemann surface 
 of Vf7, the following figure: 
 
LECTURE II 
 
 23 
 
 where we have shaded the positive half-sheets of 
 the surface and have marked the segments, 6165, 
 6se„ by heavy lines. 
 
 The points, e,, e^, e^, e„, are called the "branch 
 points" and the segments, 6162, e^e^, the "branch 
 lines" of the surface. 
 
 To construct in the i-plane the figure which is 
 the conformal representation of this Eiemann 
 surface, we conceive of this surface as cut into 
 four half-sheets, by an incision made all along 
 the real axis, and seek first the conformal repre- 
 sentation of the upper half-sheet. To obtain this, 
 we cause the point u to move, in the positive 
 sense, along the real axis, from gj through e^e^e^, 
 back (from the left) to e^ and study the corre- 
 sponding changes of value of t by means of the 
 integral, * = ( — ^ , by which it is defined. 
 
 We thus find that as the point u traces out the 
 real axis in its plane, the corresponding t traces 
 out a rectangle in its plane, which we may rep- 
 resent by the figure: 
 
 f-7/ 
 
 Fie. 2. 
 
24 MOTION OF THE TOP 
 
 to the angular points of which we have attached 
 the values of u to which they correspond, and 
 which we have shaded, since the sense in which 
 its perimeter was traced shows that it is its inte- 
 rior which corresponds to the shaded half-plane 
 of the preceding figure.* 
 
 As long as the integral which defiines t is left 
 an indefinite integral, this rectangle remains free 
 to occupy any position in the f-plane, — only the 
 directions of its sides, parallel respectively to the 
 real and imaginary axes, and their lengths, — 
 call them <«, and id^, — are completely determined. 
 But when the integral is made definite, by making 
 e„ the lower limit of integration, the angular 
 point, e„, coincides with the origin in the ^plane, 
 and the rectangle takes a definite position in the 
 plane. 
 
 From the image which we have thus obtained 
 of the one half-sheet, the images of the three 
 remaining half-sheets are to be had at once 
 by the process of "symmetrical reproduction"; 
 which yields for the Riemann surface, when 
 cut in the manner indicated, the complete 
 image : 
 
 * The figure Is a rectangle since VW is real from u = ei 
 to II = 62, and from u = es to m = e„ , and a pure imaginary 
 from « = e«, to m = ei, and from « = ea to m = 83, At et, 
 t — const, vanishes as (it — ei)i. 
 
25 
 
 The symmetry of the figure with respect to 
 the sides, e„ei, and e^e^, of the original rectangle, 
 will be at once noticed. Each of the four smaller 
 rectangles is the image of one half-sheet; the 
 shaded, of the positive half-sheets ; the non- 
 shaded, of the negative. 
 
 But we have not yet obtained the complete 
 geometrical representation of u, regarded as a 
 function of t. The Riemann surface of two sheets, 
 which we have thus far been considering, pos- 
 sesses a distinct point for every value of vTT re- 
 garded as a function of u, but not for t when so 
 regarded. The integral * is affected by an ad- 
 ditive constant if u be made to trace in the 
 Eiemann surface a closed path which surrounds 
 6162, or one which surrounds 6263, so that the 
 Riemann surface of t is one possessing the same 
 
26 MOTION OF THE TOP 
 
 branch points as the Riemann surface of y/U, 
 but having an infinite number of sheets, into 
 any one of which it is possible to move the 
 tracing point, .w, if no such cut be made in 
 the surface as that made above along the real 
 axis. 
 
 It is a great advantage of the Riemann method 
 that the complete image in the ^plane of this 
 uncut Riemann surface of an infinite number of 
 sheets may be had from the image already ob- 
 tained for the cut surface, by simply affixing a 
 rectangle, congruent with this image, to each of 
 its sides, repeating the process for the new rec- 
 tangles, and so on indefinitely, until the entire 
 ^plane is covered by congruent rectangles, any one 
 of which may be brought into coincidence with 
 any other by two translations, one in the direc- 
 tion of the real, the other in the direction of 
 the imaginary, axis. 
 
 From the result of this construction, there at 
 once follows a conclusion of the very first im- 
 portance. The image of the complete Riemann 
 surface of t entirely covers the ^plane, hut without 
 the overlapping of any of its parts. It follows im- 
 mediately, therefore, that to each point in the 
 <-plane there corresponds but a single point in 
 the Riemann surface, or that u, and ■\/U as well, 
 is a uniform function of t. 
 
LECTURE II 27 
 
 The equation connecting t and u is : 
 du 
 
 And the conckision which we have reached is, 
 that the functional relation of u with respect to t, 
 defined by this equation, is vastly more simple 
 than that of t with respect to m; to each value 
 of u there corresponded an infinite number of 
 values of t, while to each value of t there corre- 
 sponds but one value of u. As thus defined, u 
 is called an elliptic function of t. 
 
 Let ft), be the length of the side e„e^ of the 
 small rectangle, which was the image of a half- 
 sheet of the Riemann surface, and w^ the length 
 of the side e„ei; then, obviously, if we set m 
 = <^ (f), and ^0 be any point of the complete rec- 
 tangle (Fig. 3), since fo + "^i 2 o)i -|- wij 2 iw^ is for any 
 integral values of mj, m^, the corresponding point 
 of another of the rectangles, <^ (fo -I- 2 mxiai -f 2 m^iat^ 
 = <t> (to) ; or the elliptic function, u, is doubly peri- 
 odic, with the periods 2 (Oj, 2i(i,^. 
 
 Let us next consider the nature of <^ and i/' when 
 regarded as functions of t. The integrals by which 
 they are expressed in terms of u are elliptic 
 integrals of greater complexity than is the integral 
 for t. There are on the Eiemann surface of V(7 
 four points, at which each of these integrals be- 
 
28 MOTION OF THE TOP 
 
 comes logarithmically infinite ; namely, the points 
 — 1, 4-1, in the upper sheet, and the same points 
 in the lower sheet. Elliptic integrals possessing 
 such points of logarithmic discontinuity are called 
 "elliptic integrals of the third kind," and it is 
 possible to express any such integral in terms of 
 integrals of the first kind and " normal " integrals 
 of the third kind, such, namely, as possess but two 
 points of logarithmic discontinuity with the resi- 
 dues + 1 and — 1 respectively. 
 
 But if instead of making this reduction of the 
 integrals directly, we introduce those combinations 
 of <j) and \j/ which constitute our parameters a, /?, y, 8, 
 a remarkable simplification at once ensues such as 
 renders any further reduction unnecessary. Surely 
 a preestablished harmony exists between the prob- 
 lem before us, and our parameters a, /?, y, 8. 
 
 Since a = cos - • e 2 , 
 
 
 and cos • ., 
 
 2 \ 2 
 
 we have immediately 
 
 log a = J log -^— + -i^!i^=-^ 
 
 =/ 
 
 -^+i{l + n) dv, ,, - 
 2(« + l) -Vt-^'"^'' 
 
 when for c^ and f their values are substituted, 
 
LECTURE II 
 
 29 
 
 And in like manner, 
 
 log/3=J[ 
 log y =f 
 
 ■n) 
 
 lOgS: 
 
 -s 
 
 2 (m - 1) 
 
 VU+i{l-n) 
 2 {u - 1) 
 
 VU-i(l + n) 
 
 du 
 du 
 
 i log 2, 
 
 i log 2, 
 
 du 1 1 o 
 -^-Uog2, 
 
 2{u + l) 
 
 and these are all normal integrals of the third kind, 
 each with but two points of logarithmic discontinuity 
 which are distributed in the rectangle of the periods 
 as indicated in the accompanying figure, if we sup- 
 
 pose as we shall find it convenient to do later on 
 that I is less than n. 
 
 For U'=-(Z + w)2 ^yjjgj^ u = -\, and U'= 
 — (Z — rif when w = + 1. If, therefore, of the two 
 
30 MOTION OF THE TOP 
 
 values of yfU, which correspond to tt = — 1, we 
 take ((Z+h), the factor V f7 +'•(/ + m) in the nu- 
 merator of the expression for log a will be canceled 
 by the factor 2V(7in the denominator, while if we 
 take —i{}-\-n), the numerator vanishes; so that 
 the point — 1 in one of the sheets of the Riemann 
 surface of VfTis the only finite point of discontinu- 
 ity of the integral log «. It is, moreover, a logarith- 
 mic discontinuity with the residue 1, since logce there 
 becomes oo as log (m -|- 1). On the other hand, for 
 u=-jj, i.e. at e«i, log a becomes infinite as ^ log?/. This 
 again is a logarithmic discontinuity, with the residue 
 — 1, since e„ is at infinity and a branch point. And 
 like considerations apply to the remaining integrals. 
 
 By the introduction of the parameters a, /8, y, 8, 
 therefore, the four logarithmic discontinuities of 
 the integrals </», i^, are assigned one to each of the 
 four normal integrals log a, log /8, log y, log S — 
 normal integrals whose remaining points of discon- 
 tinuity, corresponding to &», coincide at the origin. 
 
 While log «, log p, log y, log 8, as now defined are 
 much simpler functions of u, and therefore of t, than 
 are <j> and i/i, their exponentials a, B, y, 8 are simpler 
 still. These are uniform functions of t having each 
 one null-point and one oo-point in every parallelo- 
 gram of periods. Such functions may always be ex- 
 pressed, apart from an exponential factor, by the 
 (quotient of two ''t- or two o--functions of the sim- 
 
LECTURE II 31 
 
 plest kind — functions which possess one null-point 
 in each parallelogram of periods, but no oo-point. 
 
 Of the i?-functions we shall only pause to re- 
 mark that Jacobi introduced them into analysis as 
 being the simplest elements out of which the elliptic 
 functions could be constructed. He obtained for 
 them expressions in the form of infinite products 
 and infinite series. They are affected by an expo- 
 nential factor when the argument is increased by a 
 period, but remain otherwise unchanged. The &- 
 functions of the simplest class, with which alone 
 we are concerned, vanish when the argument takes 
 the value zero or a congruent value. 
 
 The cr-function of Weierstrass is a more elegant 
 function of the same character. 
 
 Inasmuch, therefore, as a, p, y, 8, are functions 
 of t, which vanish for t = — ia, wi -f- ib, wi — ib, + ia 
 respectively (the values of t corresponding to the 
 points M = ± 1 in the above figure) and which all 
 become infinite for i = 0, we have for them the 
 following expressions : 
 
 o-(i) cr{t) 
 
 ..A-t ^(t-,^, + ib) ., <T(t-ia) 
 
 y-Jc,e ^^^^ , 6-ic,e ^^^^ , 
 
 where fc,., X,- are constants to be determined from 
 the initial conditions of the motion. Their values 
 
32 MOTION OF THE TOP 
 
 depend on those of the " transcendental " constants 
 (Oi, 0)2, a, 6, as the values of these in turn depend 
 on those of the " algebraic " constants, P, h, I, n. 
 
 We shall call functions such as a, 13, y, S, which 
 miss being doubly periodic by an exponential fac- 
 tor only, "multiplicative elliptic functions." All 
 elliptic functions are expressible as quotients of &- 
 or o--functions, and evidently of such quotients the 
 simplest possible are those which have a single i? or 
 <r of the simplest kind in both numerator and de- 
 nominator. We may therefore state the result of 
 our discussion in these terms : We Jiave shown that 
 our parameters a, j3, y, 8 are multiplicative elliptic 
 functions of the simplest hind, so that by introducing 
 them we have resolved the problem of the top into its 
 simplest elements. 
 
 Prom these expressions for a, p, y, 8 one may 
 obtain expressions for the nine direction cosines 
 a, 6, c • • • in the form of quotients of a-- or i9-f unc- 
 tions — such as Jaeobi got for them — with the 
 least possible reckoning.* 
 
 * Hess has remarked, in his paper on the gyroscope (Math: 
 Ann. xxix., 1887) that the quaternion expressions for the 
 nine direction cosines are very simple, and our parameters 
 are but linear combinations of the quaternion parameters. 
 Hess, however, makes no direct use of our parameters and 
 
 probably was not aware of the formula, f = "*" ^ , which 
 lies at the basis of our discussion. 7^ + » 
 
LECTUEE III 
 
 In the lecture of yesterday we reached the con- 
 clusion that our parameters a, fi, y, 8 may be 
 expressed as quotients of simple tr-functions of 
 the time t, and we now turn to the geometrical 
 interpretation of these formulas. 
 
 As I have already asked you to notice, a, p, y, 8 
 are not ordinary elliptic functions of t, but func- 
 tions which are affected by an exponential factor 
 when i is increased by a period; in consequence 
 of which I called them "multiplicative elliptic 
 functions." When t is increased by the period 
 2 0)1, they are affected by an imaginary factor of 
 the form e'"'', and when t is increased by the period 
 2 1(1)2, by a real factor of the form k. 
 
 Let us first of all consider the curve described 
 by the apex of the top on the fixed sphere. This is 
 the point Z = oo of the movable sphere, so that, 
 reverting to the formula : 
 
 «Z + ^ 
 ^ ~ yZ -I- 8' 
 
 it is obvious that the equation of the curve is 
 
 a 
 
 Z; = :^ = A;e« 
 
 a-{t + ia) 
 
 cr{t — 0)1 -|- ib) 
 33 
 
34 MOTION OF THE TOP 
 
 Like a, /8, y, 8, this ^ is defined in terms of t by 
 a multiplicative elliptic function of the first degree, 
 involving besides the exponential factor only the 
 quotient of two simple tr-functions. 
 
 This is an essential simplification of the repre- 
 sentations of this motion given hitherto. Thus, 
 were one to apply the methods used by Hermite in 
 his Applications des fonctions elliptiques, published 
 twenty years ago, and start not from the equation of 
 I in terms of Z, but from those of x+iy, —z, —x+iy 
 in terms of X+iY, — Z, — X + iY {see page 8), 
 one would obtain for the motion of the apex of the 
 top (whose coordinates are 0, 0, 1), the equation 
 
 x + iy = — 2 a/3, 
 
 which represents the motion by means of a multi- 
 plicative elliptic function of the second order. The 
 curve thus defined is not the curve traced by the 
 apex on the fixed sphere, but the orthogonal pro- 
 jection of this curve on the a^z-plane. 
 
 I shall, for convenience, call curves like those 
 which w^e have just been considering "multipli- 
 cative elliptic curves," distinguishing when neces- 
 sary between those on the sphere and those on 
 the plane, and assigning to them a degree corre- 
 sponding to the number of simple a-quotients in 
 the expressions which define them. Thus the 
 curve traced by the apex of the top on the fixed 
 
LECTURE III 35 
 
 sphere is a multiplicative elliptic curve of th.e first 
 degree, its orthogonal projection cne of the second 
 degree. The earliest example of such a curve of 
 the fiLrst degree is the herpolhode of a Poinsot 
 motion, the motion of a body about its centre of 
 gravity. That this herpolhode is such a curve 
 was first shown by Jacobi.* 
 
 It is easy to get a notion of the geometrical 
 character of the curve traced by the apex of the 
 top. For the particular case when Z — nej = 0, the 
 stereographic projection of the curve has the shape 
 indicated in the following figure : 
 
 FlQ. 5. 
 
 * Concerning the multiplicative elliptic curves, see Miss 
 Winston's dissertation : Ueher den Hermiteschen Fall der 
 Lameschen Viffereniialgleichung, GBttingen, 1897. 
 
36 MOTION OB" THE TOP 
 
 As we wish to restrict t to real values, we here 
 make ei the lower limit of integration of the 
 
 integral t = | -^-!^, or what comes to the same 
 
 thing, suppose the t of the preceding formulas 
 replaced by t' = t + iu)^. 
 
 The radius of the circle marked u = e^ is the 
 modulus of those points ^ for which t is 0, 
 2a)„ •••; for all these points u = e^. On the other 
 hand, the radius of the circle marked u — e.^ is 
 the modulus of those points t, for which t = coi, 
 3<oi, •••. 
 
 The curve of the figure is that traced by the 
 stereographic projection of ^ as i varies through 
 real values, and consists of an infinite number of 
 congruent arcs which touch the inner circle and 
 form cusps at the outer one. If the top be given 
 an initial thrust sideways (when I — 7162 is no longer 
 0), these cusps will be replaced by loops or wave- 
 crests. 
 
 Evidently any one of these arcs may be brought 
 into coincidence with the consecutive one by one 
 and the same rotation about the origin. The 
 transformation which effects this rotation is 
 tj = e'*"^, so that the meaning of the imaginary 
 factor e'*", by which ^ is affected when { is in- 
 creased by the real period 2 w^ is perfectly obvious. 
 We shall find that since f is affected by the real 
 
LECTURE III 37 
 
 * 
 
 factor K when t is increased by the imaginary- 
 period 2 iiD2, the effect on the curve of this increase 
 in t is to transform it into a curve similar to 
 itself, and symmetrically placed with respect to 
 the origin. 
 
 But before attempting a more minute examina- 
 tion of the curve traced by the apex of the top, 
 let us consider the polhode and herpolhode of the 
 motion. 
 
 On each instantaneous axis of rotation let a 
 segment be measured from the fixed point, equal 
 in sense and magnitude to the amount of rota- 
 tion about this "axis. The aggregate of these 
 segments constitute a portion of one cone if they 
 be caused to remain fixed in the moving body, 
 of another if they be caused to remain fixed in 
 space. The first cone, or the curve in which its 
 elements terminate, is called the "polhode,'' the 
 second the " herpolh'ode," and it is evident that 
 the motion of the body may be had by rolling 
 the first cone or curve on the second cone or 
 curve. 
 
 To obtain the equation of the polhode, con- 
 sider the infinitesimal rotation in time dt about 
 the axis for which the components of rota- 
 tion with respect to X, Y, Z, are p, q, r, respec- 
 tively. The axis is for the instant fixed in 
 space, and we have for the effect of the rota- 
 
38 MOTION OF THE TOP 
 
 tion on any point of the moving sphere the 
 
 equations : 
 
 X=+X' -rdtT + qdtZ', 
 T= + rdtX' + Y' -pdtZ', 
 Z = - qdiX' + pdt T + Z', 
 
 For this motion therefore the quaternion param- 
 eters (see page 4) are : 
 
 J.' = |df, B' = ^dt, G'='^dt, D'=l, 
 
 and therefore the corresponding parameters a, /3, y, 8 
 are: 
 
 a'=.l+%dt, I3' = =^dt, 
 
 y' = t^dt, 8' = l-|d. 
 
 If therefore a, ft, y, S (unprimed) be the parame- 
 ters of the transformation from the axes X, Y, Z 
 fixed in the body to the axes x, y, z fixed in space, 
 we may obtain the parameters « + da, /8 -f- dp, 
 y -f dy, 8 + c?S of the transformation which defines 
 the position of the body after the infinitesimal 
 rotation, by combining the two substitutions : 
 
 the result of which is : 
 
 ^1 = (««' + jSyOZi + («j8' -j- ;88')Z2, 
 4 = {ya< + 8y')Zx +(y|8' + SS^Z^. 
 
LECTURE III 39 
 
 It follows, therefore, that 
 
 a + da = aa' + fiy', ^ + dj3 = a/3' + /38', 
 y + dy = ya' + 8y', 8 + dS = yP' + 88', 
 
 whence dtt = (-^a+ '^ ^^ jijdt, 
 
 dp 
 
 dy=.(^ly+^-+^8)dt, 
 
 ds=l^jipiy-py. 
 
 Whence finally : 
 
 r,./ dS dp 
 
 dt ' dty 
 
 We will not stop to deriTe the corresponding 
 equations for the components tt, k, p of the herpol- 
 hode. They differ from those just obtained for 
 p, q, r only in having a and 8 interchanged and the 
 signs of p and y changed. 
 
 But I wish to make two remarks which are sug- 
 gested by the above reckoning, with regard to the 
 usefulness of our parameters a, p, y, 8. The one is 
 that two linear substitutions in terms of them 
 combine binarily instead of (juaternarily as do the 
 
40 MOTION OF THE TOP 
 
 corresponding quaternion substitutions ; the other, 
 that the four linear differential equations which 
 defiae them in terms of t, p, q, r break up into two 
 pairs, in one of which only a and ^8 are involved, in 
 the other only y and 8. To appreciate how impor- 
 tant this advantage is, one need only compare with 
 our discussion the discussion of the same question in 
 Darboux's Lemons sur la thiorie ghiirale des surfaces. 
 Returning to our spherical top, and substituting 
 in the general equations which we have Just obtained 
 the values of a, p, y, 8 which characterize its motion, 
 we have for its polhode and herpolhode not equa- 
 tions of the second degree, as was to have been 
 expected from the expressions for p + iq, etc. in 
 terms of «, /3, y, S, but much simpler expressions. 
 I cannot give the reckoning which leads to them 
 since I have not given the values of the constants 
 ^1, X„ ••■ which appear in the formulas on page 31. 
 But the expressions themselves are of the form 
 
 7 ( i-i <^(* + "1 — *'« — 46) 
 p + iq=z fc'e^' -^ ^—^ i, r = n; 
 
 <T{t) 
 
 Both the polhode and the herpolhode of the spheri- 
 cal top are elliptic plane curves of the first degree. 
 Darboux has given this result in his edition of 
 Pespeyrous' Mechanics, obtaining it by the use of 
 
LECTURE III 41 
 
 elliptic integrals instead of elliptic functions. He 
 does not call the curves elliptic curves of the first 
 degree, but curves of the same character as the 
 herpolhode of a Poinsot motion. It should be 
 added that the curves are of the first degree in the 
 case of the spherical top only. 
 
 Our theorem is closely connected with the cele- 
 brated theorem of Jacobi already mentioned : that 
 the motion of the top may be represented by the 
 relative motion of two Poinsot motions (or rotations 
 about the centre of gravity) ; for both the polhode 
 and herpolhode of the top's motion are themselves 
 herpolhodes of Poinsot motions, being elliptic 
 curves of the first degree. One may demonstrate 
 Jacobi's theorem most simply by expressing the 
 a, /3, y, S of each of the Poinsot motions in terms of 
 t, and then combining the two motions. 
 
 I may finish this part of my discussion with the 
 remark that the attention of students of the geome- 
 try of Salmon and Clebsch is apt to be confined too 
 exclusively to algebraic curves. We have before 
 us an illustration of the value of transcendental 
 curves. It is only in the very exceptional case 
 when the multiplicative factor k = 1 , and ipa is com- 
 mensurable with TT, that the curves we have been 
 studying become algebraic. 
 
 To sum up the conclusions which we have thus 
 far established ; we have proved that the motion of 
 
42 MOTION OF THE TOP 
 
 the spherical top on a fixed point of support may be 
 completely clefuied geometrically in terms of elliptic 
 curves of the first degree. We have also shown that 
 the variation of the parameters a, /?, y, 8 with the time 
 t may be pictured by curves of the same character. 
 
 Let us now resume the study of the curve traced 
 by the apex of the top. 
 
 The parameters a, /?, y, 8, and ^ are all elliptic 
 functions of the argument t, and the full meaning 
 of elliptic functions comes to light only w^lien the 
 argument is supposed capable of taking complex 
 values. Thus only, in particular, will the double 
 periodicity of the functions come into evidence. 
 There exists, then, an analytical necessity, so to 
 speak, that we complete our geometrical study of 
 the top's motion by extending it to complex values 
 of t. AVhen that has been accomplished, I shall 
 show that to the entire aggregate of possible 
 motions of the top in complex time there corre- 
 sponds the free motion of a certain rigid body in 
 non-Euclidean space, and thus bring to a definite 
 outcome the considerations which I presented at 
 the close of my first lecture. 
 
 Our problem being to determine the path traced 
 by the point ^ when the point t is made to describe 
 any path in the f-plane, it is clearly of prime impor- 
 tance that we determine first of all the image on 
 the {-sphere of a parallelogram of periods iu the 
 
LBCTCBE III 
 
 43 
 
 i-plane. To that, indeed, we shall confine our 
 attention. Instead, however, of finding this image 
 directly we shall find it easier to obtain the images of 
 the four half-sheets of the Eiemann surface of V U, 
 of which, it will be remembered, the four smaller 
 rectangles into which the entire parallelogram of 
 periods was subdivided were severally the images. 
 Let us first reproduce (in Fig. 6) the figure of the 
 parallelogram of periods (see page 29) and that of 
 the Riemann surface of Vt/i 
 
 I have given different markings to all four rec- 
 tangles in order to be able to distinguish readily 
 between their several images in the figure which 
 
44 MOTION OF THE TOP 
 
 we are to construct. It will be remembered (see 
 page 28) that log a and log -y became infinite at the 
 points M = — 1 and ?t = + 1, respectively, of one 
 sheet of the V C/"-surf ace, and that log/3 and log 8 
 became infinite at the corresponding points of the 
 other sheet — -the other functions in each case re- 
 maining finite. In the figure, a and d are the 
 images of the positive and negative halves of the 
 first of these sheets, and c and 6 the images of 
 the positive and negative halves of the second. 
 
 Ovir ^ is expressed in terms of u by the elliptic 
 integral of the third kind : 
 
 , > , f(t\ r'—-\/U+i(nu — l) du 
 log^ = logy=j -,-L _. 
 
 We may now draw the following conclusions 
 immediately : 
 
 — log I - ) is complex along the segments 6162, 
 
 ege„ of the real axis of the w-plane, but real along 
 
 a 
 the segments 62^3) e^ej. Therefore ~ or ^ moves 
 
 along a meridian of the Z;-sphere when u moves 
 along the real axis from gj to Sg or from e„ to 6] ; 
 but, on the other hand, describes one of the arcs 
 which appeared in the figure of the real motion of 
 the top's apex, when u moves on the real axis from 
 61 to 62, and an arc different from this, when u moves 
 from 63 to e„. 
 
LECTURE III 45 
 
 ^gain, ^ log l-j vanishes when u = e„, in the 
 
 first approximation as — , and takes the finite value 
 
 1 ** 
 
 — when II, = 62 (this because of the hypothesis 
 
 1 — 63 
 
 which we retain here, that l — ne2 = 0); when rt=ei 
 or 63, on the other hand, it becomes infinite, as 
 (u — 61)"* or (u — 63) 2. Therefore the curve traced 
 by the point ^ as the point u moves along the real 
 axis from e„ through e^, 63, 63, to e„ will present 
 angles whose measure is tt at the points corre- 
 sponding to e„ and 63, and angles whose measure 
 is ^ at the points corresponding to ej and eg. 
 
 I will not give the image of the VT/'-surface on 
 the sphere, but the stereographic projection of this 
 image on the xy-T^la,ne from the point ^ = 00. If to 
 the explanations already given it be added that ^ 
 
 whose value in terms of t is fee''' /^^ — ^ — , be- 
 
 a-(t — 0)1 -)- ib) 
 
 comes and 00 respectively at the points — 1 and 
 + 1 of the contour of the half -sheet or rectangle a, 
 and remains finite and different from for all 
 points on the contour of &, it will readily be seen 
 that the images of the half-sheets or rectangles a, 
 b, are roughly of the form indicated in the follow- 
 ing figure : the two contours which we have marked 
 e„ ej 63 63 e„ being the stereographic projections 
 of images of the real w-axis first when this axis is 
 
46 
 
 MOTION OF THE TOP 
 
 regarded as the contour of the positive half-sheet 
 a, second when it is regarded as the contour of 
 the negative half-sheet 6. 
 
 The two arcs ejej are similar 
 metrically placed with respect 
 point ^ = 0. The one which lii 
 the left appeared in the figure 
 the real motion of the top's apex 
 (Fig. 5). 
 
 If now we complete th 
 figure by a second half 
 symmetrical with this 
 first half with re- 
 spect to the 
 horizontal 
 axis fje^, we 
 obtain the 
 image of 
 the en- 
 tire 
 
 Fis. 7. 
 
LECTURE III 47 
 
 ' C/^surf ace or of tlie entire parallelogram of periods 
 in the f-plane (Fig. 8). We suppose an incision 
 made in the V [/-surface along the segment 616263 of 
 the real axis. 
 
 It will be noticed that the image covers doubly 
 the portion of the plane -which lies within the 
 two arcs e^, e^, 63, ^ = 00 , which lie to the right, 
 the two sheets being joined along a branch line 
 which runs from e„ to ^ = 00 . From the figure 
 we infer that e„ is a branch point of t, but not so the 
 point ^=00 ; for a circuit cannot be made of the point 
 ^ = CO without passing into the portion of the plane 
 bounded by the half -ares e^, e,^, ([ = 0, lying to the 
 left, which does not belong to the image. And these 
 conclusions may readily be verified by reckoning. 
 
 We may describe our figure as a quadrilateral, one 
 of whose pairs of opposite sides are the rectilineal 
 segments running from the points eg, through ^=00, 
 and which, were they produced, would intersect at ^=0, 
 and the other pair, the two curvilinear arcs 626162. 
 
 The sides of each pair go over into each other 
 by the substitution of ^ which corresponds to a 
 change of t by one of the periods 2o)j, 2ia)2: the 
 straight sides by the rotation about ^ = defined 
 by the " elliptic substitution " ^' = e'*o^, which we 
 have already considered and which we have indi- 
 cated in the figure by the double-headed curved 
 arrows; the curved sides by the transformation 
 
48 
 
 MOTION OF .THE TOP 
 
 Fio. 8. 
 
LECTUBE m 49 
 
 defined by the " hyperbolic substitution " ^' = k^, 
 in consequence of which they are similar and sym- 
 metrically placed with respect to the centre of 
 similitude ^ = 0. In the figu.re we have indicated 
 the latter transformation by the double-headed 
 straight arrows which intersect at ^ = 0. The sig- 
 nificance of both the periods 2 <0], 2 iu>.2 for the curve 
 traced by the apex of the top is thus made evident 
 by our figure. And indeed we have now clearly 
 before us for the first time the reason that the 
 curve described in real time should be represented 
 by elliptic functions. It is but a portion of the 
 complete curve, or rather domain, which comes to 
 light when we avail ourselves of the entire field of 
 complex numbers in which the representation of 
 both periods is alone possible. 
 
 The Eiemann surface determined by ^ = ^^, the 
 
 curve traced by the opposite extremity of the top's 
 axis, Z = 0, may be constructed similarly. 
 
 For real values of t we have ^-y- = — ^Vf > which 
 
 8(f) a(t) 
 
 means simply that ^-^ and ^-i are opposite ex- 
 y{t) 8{t) 
 
 tremities of one and the same diameter of the 
 sphere. For complex values of t this formula is 
 to be replaced by the more general one 
 
 7(0 «(0 
 
60 MOTION OF THE TOP 
 
 If now we suppose these two Eiemann surfaces 
 to be projected back again to the surface of the 
 fixed sphere, and the points of the two which 
 correspond to the same value of t to be joined, 
 the resulting system of rays will represent the oo^ 
 positions which the axis of the top may take in 
 the general (non-Euclidean) motion which cori-e- 
 sponds to any motion of t in the parallelogram of 
 periods. 
 
 Of these co^ "axes," only those pass through 
 the centre of the sphere which correspond to real 
 values of t. These are the axes which meet the 
 curved arc 626162 of the preceding figure which lies 
 to the left. Those axes which meet the other 
 curved arc 626163 intersect in another point of the 
 central line (i.e. of the vertical through the centre 
 of the sphere); namely, the point into which the 
 centre of the sphere is transformed by the hyper- 
 bolic substitution already explained. A visible 
 representation of the possible motions of the top's 
 axis in complex time is to be had by constructing 
 
 the figures for - and ^ on an actual sphere and 
 
 Y S 
 
 joining a number of corresponding points by 
 straight lines. 
 
 The doubly infinite systems of the rays which 
 are elements of the polhodes and herpolhodes of 
 all motions possible in complex time, may be con- 
 
LECTUEB III 51 
 
 structed in like manner, and a complete geomet- 
 rical representation be thus obtained of the top's 
 motion. The constructions are more complicated, 
 but there is no essential difficulty in carrying 
 them out. 
 
 In fact, the only serious difficulty in this entire 
 method of discussion is, that all our ordinary con- 
 ceptions of mechanics involve the notion that time 
 is capable of but one sort of variation. We are so 
 accustomed to regard the mechanical conditions 
 which correspond to small values of *, as, so to 
 speak, the cause of those which correspond to 
 greater values, and to picture the changes of con- 
 figuration as following one another in definite order 
 with the varying time, that we find ourselves at a 
 loss for a mechanical representation when t, by 
 being supposed complex, becomes capable of two 
 degrees of variation. 
 
 To avoid this difficulty as far as possible, let us 
 suppose t no longer capable of varying in every 
 direction in the parallelogram of periods, but only 
 along a line parallel to the real axis. In other 
 words, in t = ti + it^, let us regard t^ as constant in 
 each particular case, and t-i as alone varying. In this 
 manner, by subsequently giving t^ all possible values, 
 we may take into account all possible complex 
 values of t, but we conceive them as ranged along 
 the oo* parallels to the real axis. Eegarded thus, 
 
52 MOTION OF THE TOP 
 
 « B 
 
 the Riemann surfaces -, ^ become carriers of cer- 
 y 
 
 tain curve systems, and the system of oo^ axes is 
 distributed among oo^ ruled surfaces. 
 
 In this manner we separate the totality of the 
 positions of the top in complex time into an infi- 
 nite number of simply infinite sets of positions. 
 These sets of positions are characterized not only 
 by the initial values of t, but by the values of the 
 constants of integration, which must have been 
 introduced had the reckoning Avhich we have merely 
 sketched been actually carried out. It should per- 
 haps have been stated earlier that in the interest 
 of complete generality these constants must now 
 be supposed complex, for we are now operating 
 in the domain of complex numbers. Moreover, 
 only by supposing them complex shall we have 
 constants enough at our disposal to meet all the 
 conditions of our generalized problem of motion. 
 
 So far our figures have been constructed with a 
 view to obtaining a clear geometrical representation 
 of the entire content of our analytical formulas. 
 But their chief interest lies in this : that one can 
 give them a reed dynamical meaning, that one 
 can find a real mechanical system by whose mo- 
 tions they may be generated. I assert that one can 
 determine a certain free mechanical system, namely, 
 a rigid body freely moving in non-Euclidean space 
 
LECTURE III 53 
 
 under the action of certain definite forces, which in 
 real time carries out exactly that infinity of forms of 
 motion which we have just been describing, the one 
 or other of them according to the choice made of the 
 initial conditions of motion. The mechanical sys- 
 tem is a generalized one, but it belongs to tlie do- 
 main of real dynamics. 
 
 Let us consider the general problem of the motion 
 of a rigid body under the action of any forces, in 
 the non-Euclidian space whose absolute is the sur- 
 face : 
 
 a^ + 2/' + «'-«' = 0. 
 
 The earliest investigation of the motion of a rigid 
 body in non-Euclidean space was made by Clifford 
 in 1874 — though the investigation was not pub- 
 lished until after his death, in his collected works. 
 The same problem has been considered also by 
 Heath in the Philosophical Ti-ansactions, 1884. Both 
 these mathematicians, however, have treated the 
 case of the elliptic non-Euclidean geometry, not 
 the hyperbolic, and have contented themselves 
 with establishing the differential equations of the 
 problem. 
 
 I shall proceed analytically, as this method is 
 more readily imderstood by one who is not well 
 versed in non-Euclidean geometry, and immediately 
 obtain differential equations for the motion of a 
 certain rigid body in non-Euclidean space perfectly 
 
54 MOTION OP THE TOP 
 
 analogous to the equations for the motion of the 
 top in real time, but involving two sets of variar 
 bias. 
 
 To have the general case before us at once, I sup- 
 pose the parameters <j), \f/, '?, and the time t, all com- 
 plex and set 
 
 •^ = <^i + i4>i, "A = i/'i + i<pi, '' = ''i + i'%, t = ti + iti. 
 
 These parameters are connected with T and V, 
 the kinetic and potential energy, by the well-known 
 Lagrange equations : 
 
 rfHi - - <g:) 
 
 w) ^ 8(r- V) "V8<^V _ 8(r-F) 
 at ~ u ' dt ~ S(^ ' 
 
 dt 8i^ 
 
 In these equations set 
 
 Since, then, ^ = ^ + {1^' 
 
 and similarly. 
 
 .82j!_ -Sli. 
 8»\ *S*'j' 
 
 8^' S4,\ S^'s 8^' 8i/.'i 8i/-'j' 
 
LECTURE III 55 
 
 and since, furthermore, by our hypothesis, dt = dti, 
 the first of our equations breaks up into the two 
 equations involving real variables only, 
 
 \U'J _ 8(T,-V,) V8.»'J _ 8(ri-F.) . 
 d«i S*i ' dti S'l'a ' 
 
 and the remaining two equations behave similarly. 
 
 Thus, every real mechanical problem again re- 
 duces to a real problem when the variables are 
 made complex, provided the real part only of the 
 complex t be supposed to vary, but the problem of 
 a motion involving twice the number of variables. 
 
 Applying this general conclusion to the particular 
 question before us, it is evident without any further 
 discussion that the problem of the motion in com- 
 plex time of a top whose point of support is fixed is 
 changed into a problem of real dynamics ; the 
 problem of the non-Euclidean motion of a rigid 
 body. This motion has six degrees of freedom in- 
 stead of three, corresponding to the six parameters, 
 I?!, i^j, 4>j, 4)2, i/'i, </'2, and its kinetic and potential 
 energy are Tj and Vi, the real parts of the complex 
 T and r. 
 
 But what is the rigid body, and what the force 
 producing the motion ? We shall content ourselves 
 with simply answering these questions without en- 
 tering upon the considerations appertaining to non- 
 
56 MOTION OF THE TOP 
 
 Euclidean geometry by which our conclusions are 
 reached. 
 
 The equation of the absolute being 
 
 9? + y'^ + z^-t'^ = 0, 
 the integral 
 
 S, 
 
 (ux + vy + ivz + latf ,^ 
 
 {,1^ ^ v" + w" - J'){x'' + f- + z^ - f) ' 
 
 evaluated throughout any body in the correspond- 
 iug non-Euclidean space, is called the " second 
 moment" of the body with respect to the plane 
 whose co-ordinates are u, v, w, w. In the particular 
 case before us this integral, lohen evaluated, will he 
 equal to 1, independently of the values of n, v, w, o). 
 
 Remembering that u, v, w, <o are constants with 
 respect to the integration, the result may be written 
 
 i ,'^2 I "2 "i > '"'l^icli therefore = 1. 
 
 w + v' + W^ — to' 
 
 Now the surface whose equation in tangential co- 
 ordinates is 
 
 Aii? + 2Buv-\ =0 
 
 is called the " null-surface." In the case before us, 
 therefore, the null-surface coincides with the absolute. 
 This is the rigid body of our non-Euclidean motion. 
 The force producing the motion may be defined 
 as follows : In the figure (Fig. 9) let g represent the 
 fixed axis of gravitation (through the point of support 
 
LECTURE in 57 
 
 of the top), r the axis of the top, and p the non- 
 Euclidean perpendicular common to g and ?•. The 
 angle between g and }• is then 
 defined as * = >9i + p%, where i?i 
 represents the angle between the 
 planes gp and rp, and v^^ in non- 
 Euclidean angular measure is the 
 distance p. 
 
 The force is then the wrench 
 
 , . . . , . Fig. 9. 
 
 represented in intensity by P sm fl, 
 of which the real part represents the rotating force 
 acting about p and the imaginary part represents 
 the thrust along p. 
 
 In conclusion, allow me to remark once again 
 that this non-Euclidean geometry involves no meta^ 
 physical consid-eration, however interesting such 
 considerations may be. It is simply a geometrical 
 theory which groups together certain geometrical rela- 
 tions in reed space in a manner peculiarly adapted to 
 their study. 
 
LECTURE IV 
 
 In the latter part of yesterday's lecture we ven- 
 tured a little way into what Professor Newcomb 
 has called the " fairyland of mathematics." Ignor- 
 ing the limitation of the top's motion to real time, 
 we gave full play to our purely mathematical 
 curiosity. And there can be no doubt that it 
 is proper and indeed necessary within due limits 
 to proceed after this manner in all such investiga- 
 tions as that now before us. It is possible only 
 thus to develop a strong and consistent mathe- 
 matical theory. But we should not yield ourselves 
 wholly to the charm of such speculations, but rather 
 control them by being ever ready to return to the 
 actual problems which nature herself proposes. 
 
 We turn again to-day, therefore, to the real top, 
 and proceed to investigate its motion when the 
 point of support is no longer fixed, but movable 
 in the horizontal plane. This is the case of the 
 ordinary toy top. 
 
 It has been well known since the time of Poisson 
 that the differential equations of this motion can 
 be integrated in terms of the hyperelliptic inte- 
 58 
 
LECTURE IV 59 
 
 grals. And it is the main purpose of my present 
 lecture to show that these integrals may be treated 
 in a manner quite analogous to that in which the 
 elliptic integrals were treated, by aid of the general 
 " automoijihic functions," of which the elliptic func- 
 tions are a special class. 
 
 The "toy top" has five degrees of freedom of 
 motion, two of them relating to the horizontal dis- 
 placement of the centre of gravity, and the other 
 three to the motion around this centre. The hori- 
 zontal motion of the centre of gravity is very 
 simple, being, as is well known, a rectilinear mo- 
 tion of constant velocity. Consequently, no essen- 
 tial restriction of the problem is involved in 
 assuming the horizontal projection of the centre 
 of gravity to be a fixed point. By this assump- 
 tion the problem is again reduced to one of three 
 degrees of freedom only, and we have besides t no 
 other variables to consider than the parameters 
 <^, i/r, '? or «, j8, y, S of the previous discussion ^ — 
 the parameters here defining the position of the top 
 with respect to axes through its centre of gravity. 
 
 To obtain first the ordinary formulas which de- 
 fine the motion in terms of the astronomical param- 
 eters : let G represent the weight of the top, s the 
 distance of its centre of gravity from the point of 
 support, and again represent the product Os, i.e. the 
 static momentj by P. Also, for the sake of sim- 
 
60 MOTION OF THE TOP 
 
 plicity, let us again suppose that the three princi- 
 pal moments of inertia of the top, in this case 
 with respect to the axes through its centre of 
 gravity, are all equal to 1. 
 
 Th^n the kinetic energy, T, and the potential en- 
 ergy, V, are given by the ifollowing equations : viz. 
 
 T= i{<l>" + >li" + 2 ^V' cos .'> +'*'^ + Ps sin^ » ■ i?'^), 
 
 F=Pcos.9, ' 
 
 which differ from the corresponding expressions 
 in the special case where the point of support is 
 fixed only in the appearance of the additional term 
 Ps ■ sin^* • *'^ in T. As this term will disappear if 
 s = 0, though we take Gs, i.e. P, different from zero, 
 the elementary case may be described from the 
 present point of view as that of a top of infinite 
 weight whose centre of gravity coincides with its 
 point of support. 
 
 On substituting these values for T and V in the 
 first two Lagrange equations, 
 
 dt ' dt 
 
 We obtain ilnhiediately, as before, the two algebraic 
 
 first integrals 
 
 d>' + \j/' co^ 7? =? »>, 
 
 ip' + ^' cos * = I. 
 
LECTURE IV 61 
 
 If from these last equations we reckon out <^' 
 and i/j', and substitute the resulting values in the 
 integral of energy 
 
 T+V=h, 
 
 we obtain t, </>, and tj/ in thje foi;m of integrals in 
 terms of the variable &. 
 
 As before, we set u = cos r% and 
 
 U= 2 Pit' -2 hu^ + 2 {In dip^u+{2h-P- n% 
 
 when these integrals become 
 
 -i 
 
 •dwV(l + Ps) - Psu^ 
 
 vi r 
 
 n — lu dw V(l + Ps) — Psu? 
 
 ri — nu du^(l + Ps) - Ps«^ 
 "^ J 1 - M^' ■ V^ ' 
 
 These formulas differ from the corresponding 
 formulas for the elementary cage in tjiat the new 
 irrational factor V(l + Ps) — Psu^, here appears in 
 the numerator of each integrand. ,In consequence^ 
 we have now to do with hyperelliptic integrals', 
 p'= 2.' In 'addition to thp former branch-points of 
 the Riemalnn Surface in the ^i-plane, viz. ej, 62, 63, e„, 
 two new real branch-points appear, viz. : 
 
 .=.Vi 
 
 + Ps 
 Ps 
 
62 
 
 MOTION OF THE TOP 
 
 I shall call them e^, eg, and assume them to be nu- 
 merically greater than 63. The Eiemann surface 
 is therefore a surface of two sheets with six branch- 
 points 6], 62, 63, 64, e„, Be, ranged along the real axis 
 of the j<-plane, as indicated in the following figure : 
 
 In addition to the branch-points, I have indicated 
 the positions of the points -t- 1, — 1, since these 
 particular values of u, corresponding to >? = 0, 
 fl = TT, play, as in the elementary case, a special 
 rSle in our discussion. 
 
 The time t is no longer an integral of the first 
 kind; that is to say, an integral which remains 
 finite for all values of u, but an integral of the 
 second kind, which becomes infinite for u = oo, as 
 ■\/ —2su. An integral of the second kind, it may 
 be added, is one having a point of algebraic dis- 
 continuity only. The integrals <^ and 1//, on the 
 other hand, have each of them, as before, four 
 logarithmic points of discontinuity ; namely, the 
 four points m = ± 1 of the Riemann surface. 
 
 The first step to be taken is to replace the in- 
 tegrals <^ and i/» by normal hyperelliptic integrals 
 
LECTURE IV 63 
 
 of the third kind ; that is, by integrals possessing 
 each but two logarithmic points of discontinuity 
 with the residues + 1 and — 1. This is accom- 
 plished precisely as in the elementary case, by 
 introducing log a, log p, log y, log S. As before, 
 these prove to be normal integrals of the third 
 kind, each having a logarithmic discontinuity (with 
 the residue + 1) at one of the points m = ± 1, and 
 all having a second logarithmic discontinuity in 
 common (with the residue — 1) at the point 
 w = CO. This follows at once from the result of 
 the reckoning if it be noticed that the expression 
 (1+Ps) — Psu^ reduces to 1 for u = ±l. 
 
 It is evident, therefore, that the parameters a, 
 P, y, S play the same fundamental rdle here as in 
 the case of the top whose point of support is fixed. 
 And in the following discussion we shall no 
 longer use, <^ and i/f, but a, j3, y, S. These vari- 
 ables possess on the Eiemann surface a 0-point 
 each at one of the four points u = ±l, and a com- 
 mon oo-point at u = cc. I have not thought it 
 necessary to enter into the details of this reduc- 
 tion, as it is so completely analogous to the "re- 
 duction in the more elementary case. 
 
 But when we attempt to repeat the next step 
 of the previous discussion, and endeavor, by in- 
 verting the hyperelliptic integral t, to assign to 
 t the role of independent variable, we find at 
 
64 
 
 MOTION OI" THE TOP 
 
 once that there is a profound difference between 
 our present problem and the previous more special 
 problem. This difference is masked when wg con- 
 fine our attention to . the top's motion in, real 
 time. For as t varies, remaining always real, the 
 value of u vibrates as before between the values 
 e\ and e^, while <^ and i/r are each increased by 
 real periods. , The difference comes to light, how: 
 ever, as soon as, allowing t to take complex values, 
 we proceed to construct in the (-plane the image of 
 the Kiemann surface. As the image of a half-sheet 
 of this surface, we have now, instead of the simple 
 rectangle of the elementary case, an open hexagon 
 with one of its apgular points at infinity, as in the 
 following figure : 
 
 Pio. 11. 
 
 and when by the methods of symmetrical and con- 
 gruent reproduction, we go on to construct from 
 this figure the image of the entire Eiemann sur- 
 
66 MOTION OF THE TOP 
 
 is, perhaps, the greatest achievejnent of Jacobi, 
 and for general investigations of the highest im- 
 portance, but it promises us little aid in the prob- 
 lem which we are considering. To avail ourselves 
 of it, we should need first to develop a method 
 for determining what values of V], V2 correspond 
 to the same value of t. We are therefore reduced 
 to the direct computation of hyperelliptic integrals 
 if we wish to avoid the complicated equation for 
 Vi and V2 which results if we eliminate t. 
 
 Is it possible, then, by any means whatsoever, to 
 obtain for the general motion of the top formulas 
 analogous to those which we succeeded in establish- 
 ing for the top whose point of support was fixed ? 
 Yes, by availing ourselves of the theory of the uni- 
 form automorphic functions. 
 
 A uniform automorphic function of a single vari- 
 able ij is a function f{rj), which satisfies the func- 
 tional equation 
 
 
 where a^, b^, c,,, d^ have given constant values for 
 each of the values of v : 1, 2, 3 ■ ■ • 00 — for all of 
 which the functional equation is satisfied. 
 
 The automorphic functions, therefore, are func- 
 tions which are transformed into themselves by an 
 infinite but- discontinuous group of linear substitu- 
 
LECTURE IV 67 
 
 tions. They are the generalization of the elliptic 
 functions which consists in generalizing the perio- 
 dicity of these functions, but leaving the number 
 of the variables unchanged, while Jacobi's hyper- 
 elliptic functions are a generalization which consists 
 in increasing the number of variables, but leaving 
 the periodicity unchanged. 
 
 I shall present what I have to say regarding 
 them geometrically. And, indeed, the general no- 
 tion of these automorphic functions, as well as 
 the knowledge of their most important properties, 
 originated from geometrical considerations, and 
 geometrical considerations only. Even now the 
 analytical details of the theory have been only 
 partially developed. 
 
 Our problem, as we are now to conceive of it, is 
 this : to define a variable rj, of which t, a, j8, y, S 
 shall he uniform, automorphic functions, as were a, fi, 
 y, S of i itself in the elementary case. 
 
 To revert to the elementary case — the fact that t 
 was itself a " uniformizing " variable, i.e. a variable 
 of which u was a uniform function, was brought to 
 light by finding that when the image in the f-plane 
 of a single half-sheet of the Eiemann surface on the 
 ij-plane was reproduced by symmetry and congru- 
 ence, this image covered the i-plane simply. May 
 we not, then, construct in the plane of a variable 
 rj a rectangular hexagon which shall be the image 
 
68 MOTION OF THE TOP 
 
 in, the ly-plane of a half-plane m,, and which on 
 being reproduced shall cover the »;-plane or a 
 portion of it simply, and then subsequently, fr9m a 
 study of the conditions which determine this hexar 
 gon, derive in definite analytical form the functional 
 relation between rj and u ? 
 
 It is in fact possible, as the theory of automor- 
 phic functions shows, to construct such a rectartgu- 
 lar hexagon, and -that in essentially but one way. 
 Its sides are not line segments, biit arcs of Circles 
 which themselves cut the real axis of the 7;-plane at 
 right angles. It has the following form : 
 
 Fra. 12. 
 
 The mere geometrical requirement that the figure 
 be made up of arcs of circles which cut the real axis 
 orthogonally, and cut each other orthogonally also 
 at the six points g], e.2, e^, e^, e„, e^, is of course not 
 enough to determine it completely. There are a 
 certain number of parameters which remain unde- 
 termined, and which are to be so determined that 
 the hexagon is an actual conformal representation 
 
LECTUBE IV 69 
 
 of the half M-plaiie with the given branch-points 
 Bi, 62 ••• e^. The fundamental theorem of the theory 
 pf autoiuorphic functions declares that this can be 
 accomplished in one, and essentially but one, way. 
 
 Having determined the image of the one half- 
 sheet of the Eiemann surface on the w-plane, the 
 infinitely many reiflaiuing images are to be had 
 by constructing the figure into which the original 
 image is transformed by inversion with respect to 
 each circle of which one of its sides is an arc, by 
 repe9,ting the same construction for the resulting 
 hexagons, and so on indefinitely. 
 
 By this process the entire upper half of the 
 ij-plane is simply covered without overlapping by 
 rectangular hescagons, whose sides are circular arcs. 
 Each of these hexagons is an image of a half -sheet 
 of the Riemann surface. And if they be alternately 
 shaded and left blank, the shaded ones are linages 
 of positive half-sheets, the blank ones of negative 
 half-sheets of the surface. 
 
 Evidently, then, to a single point in the ij-plane 
 there corresponds but a single point in the Riemann 
 surface, or u and -y/U are uniform functions of rj. 
 On the other hand, the points in two of the hexa- 
 gons which correspond to the same value of u, V U, 
 and may be called " equivalent points," are con- 
 nected by a formula of the form «' = -^ p, as in 
 
70 MOTION OF THE TOP 
 
 the special elliptic case the coiTesponding points of 
 two of the parallelograms of periods were connected 
 by the formula t'=t+2 wiiwi + 2m2i<02. Thus u and 
 Vf/are uniform automorphic functions of rj, satis- 
 fying the equation : 
 
 I may remark that Lord Kelvin made use of 
 this sort of symmetrical reproduction more than 
 fifty years ago in his researches on electrostatic 
 potential. But his figures were solids bounded by 
 portions of spherical surfaces, and his aim was 
 so to determine these that only a finite number 
 of other distinct solids should result from them 
 by the process of reproduction. 
 
 Not only u and VCT, but also VI + Ps — Psu^, 
 and again t, a, /?, y, 8, arc uniform functions of 
 our new variable i;, functions, it may be added, 
 which exist only in the upper half of the r;-plane. 
 Hence -q is the uniformizing variable tvhich we have 
 been seeking, the variable which plays the rdle taken 
 by t in our discussion of the special problem. 
 
 We turn therefore to the consideration of t, a, /3, 
 y, S, regarded as functions of r/. 
 
 The variable t is affected additively by the linear 
 substitutions of rj which correspond to the suc- 
 cessive reproductions of the figure ; i.e. with every 
 substitution it is increased by a constant. More- 
 
LECTURE IV 71 
 
 over, it becomes infinite, and that simply infinite 
 algebraically, at all those points of the ij-plane 
 which correspond to the point e^ of the w-plane, 
 the points, namely, which are equivalent to the 
 single angular point marked e„ in the hexagon 
 of our figure. 
 
 On the other hand, a, ji, y, S, are affected multi- 
 plicatively by the linear substitutions of ly. Each 
 becomes zero in one series of equivalent points, and 
 that simply, and each becomes infinite, and that 
 also simply, in another series of equivalent points. 
 
 The oD-points are the same as those for which 
 t becomes infinite; the 0-points are the points on 
 the perimeters of our hexagons which correspond 
 to the four points m = ± 1 of our original Riemann 
 surface of two sheets on the it-plane. The two 
 points corresponding to ?« = + 1 we may name a', 
 a", and the two points corresponding to u=— 1, 
 b', b", in such a manner that the series of equiva- 
 lent 0-points of «, /8, y, 8, correspond respectively 
 to a', b', h", a". 
 
 On this characterization of our functions t, a, /3, y, 8, 
 we have now to base their analytical representation 
 in terms of r]. This is to be accomplished by means 
 of the functions which in this more general case of 
 the automorphic theory play the same fundamental 
 role as the elliptic o--functions in the more elemen- 
 tary case — the so-called prime-forms. The prime- 
 
72 MOTION OF THE TOP 
 
 form is not a function of rj, 'but a homogeneonx 
 
 function of the first degree of rji, tj2 (where — — rj); 
 
 like the elliptic cr-function, it vanishes at all of a cer- 
 tQiiv, series ofequ ivalent points, and is uoiohere, infinite. 
 
 I use the name prime-form because all the al- 
 gebraic integral forms belonging to the Jliemann 
 purface admit of being similarly expressed as 
 products of suitably chosen prime-forms, just as in 
 ordinary arithmetic integers as products of prime 
 pumberst It niay be added that these prime-forms 
 are not completely determined quantities. They 
 may be . altered by certain factors, the exact ex- 
 pression of which here .would cause too serious a 
 digression. 
 
 If now we represent the prime-form whose zero- 
 points are the series of equivalent points correspond- 
 ing to the point m of the Riemann surface by the 
 symbol 2(»?i, 1/2 ; "i), we have the following analytical 
 representation of pur functions tj a, /S, y, 8, viz. : 
 
 ^ ^ S(r;i,772; g') ^ „ ^ S(7;i,»;2; S') ^ 
 
 2(771, 7/2; 600)' 2(571,172; e„)' 
 
 ^ 2(171,172; b") ^ g ^ 2(171,172; a") 
 
 2(171, 172 ; e«,)' 2(171, 172 ; e„) 
 
 •, And so we find here, as before, that the function's 
 a, p, y, 8 prove to be the simplest elements for the 
 
LECTURE IV 73 
 
 representation of the top's motion. They are the 
 simplest quotients of the elementary functions of 
 the " hyperelliptic body " which has replaced the 
 ;' elliptic body " of our earlier discussion. 
 
 It may be remarked that these formulas at once 
 reduce to t = rj and the previously obtained elliptic 
 
 formulas on making the hypotheses P ^ 0, s = 0, 
 
 which are equivalent to supposing the ,, point of 
 support fixed. 
 
 Moreover, it must be said that these expres- 
 sions for t, a, ft, y, 8 are only to be understood ap 
 having a formal significance. There is altogether 
 lacking the actual determination of the constants 
 left at our disposal by tlie definitions of the S's, an<i 
 which, it may be added, differs for the different 2's 
 which appear in the formulas for t, a, fi, y, 8. 
 
 And with this we come upon the point at which this 
 theory is still incomplete. The exact determination 
 of the formulas, and in general the means of reckon- 
 ing them out by practicable methods, are for the 
 iaost part wanting. The theory of the automorphic 
 functions which for a time was a matter of principal 
 interest in the theory of functions has in recent 
 years not attracted the ' attention nor found the 
 support which it seems to deserve. I have there- 
 ■fore the more gladly laid stress here on the fact that 
 these are not only functions possessing a theoretical 
 
74 MOTION OF THE TOP 
 
 interest, but functions which necessarily present them- 
 selves if one luill completely solve even the simplest 
 problems of mechanics. 
 
 Had we the time, we should find it interesting to 
 consider the geometry of this more general case of 
 the top's motion also. 
 
 I will, however, give the equation of the curve 
 traced on the horizontal plane by the point of sup- 
 port. It is x + iy = 2 otySs, as results from the for- 
 ■mulas on page 8, by giving X, Y, Z the values of 0, 
 0, — s, respectively. For the values of x and y de- 
 pend on a, j8, y, S alone, these quantities, in the 
 present case, conditioning the motion of the centre 
 of gravity up and down its vertical, and no terms 
 appearing in the expressions for x and y due to 
 this motion. 
 
 And I may also make the general remark that in 
 this geometrical study the non-Euclidean interpreta- 
 tion plays an important r61e. For while the curves 
 traced by the apex, etc., have in real time a form 
 quite similar to that in the case of the fixed point 
 of support, the Eiemann surface as described by the 
 apex on the fixed sphere brings fully into evidence 
 the difference between the elliptic and hyperelliptic 
 characters of the two motions. Instead of the 
 quadrilateral which was represented in Fig, 8 we 
 should here have a hexagon.