ApeilS, 1898.] 



SCIENCE. 



471 



spectively, each of radius r and in congru- 

 ence, the variables Z and Z defined by the 

 ratios 



static moment of the top with respect to 

 the fixed point. Then 



C = 



X ■}- iy r -\- z 



if 



X+iY _ r + Z 

 T — Z ~ X—iY 



will be parameters each of which determines 

 a point on the fixed and movable spheres, 

 respectively. The unique advantage of 

 these non- symmetrical parameters is that 

 when the movable sphere (supposed fixed 

 in the body) rotates, the relation of the 

 parameters C and Z is a linear equation of 

 the form 



' rZ + <5 



where a and 5, y and ;9 are conjugate imagi- 

 naries. These quantities connected by the 

 equation a5 — /3^=l, together with?, are 

 used as variables specially adapted for 

 treating the top problem. Hence a scheme 

 of orthogonal substitution and a direct ex- 

 pression of the new parameters in terms of 

 the Eulerian and rotational parameters is 

 fully developed. The lecture closes with 

 an even broader interpretation of C for the 

 case when a and -/-, /S and (5 are not conju- 

 gate, and time (i) for convenience in the 

 theory of functions is also considered com- 

 plex. 



Starting on more familiar ground, the 

 second lecture begins with a direct attack 

 of the problem of rotation of a body (top) 

 about a point other than its center of mass. 

 Klein uses the expression for kinetic and 

 potential energy in terms of Eulerian speed 

 coordinates, the three corresponding La- 

 grangian equations of motion and the law 

 of the conservation of energy to reduce the 

 rotation to the following succinct specifica- 

 tions : Let *, ^, >}' be the Eulerian coordi- 

 nates and put cos '9 = u. Let [7 be a poly- 

 nomial of the third degree in it, involving 

 besides only integration constants I, n, h, 

 and the (maximum and therefore constant) 



n — lu 



/au ^11 — 



N/C7' 



/I — nu du 



so that the motion is completely given (La- 

 grange) in terms of quadratures. Unfor- 

 tunately, however, these integrals are 

 elliptic and, except in the special cases 

 worked out by Professor Greenhill, do not 

 admit of algebraic treatment, while the 

 2d and 3d integrals are, beyond this, com- 

 plex in type. Jacobi, to whom the intro- 

 duction of elliptic functions is due, was 

 thus able to make an immense stride for- 

 ward by expressing the Lagrangian inte- 

 grals u, <p. 4', and therefore the equivalent 

 cartesian direction cosines, as (one- valued) 

 >^-functions of time ; but while the direction 

 cosines thus become much simpler time 

 functions than the integrals, they are far 

 more complicated than Klein's parameters 

 a, /?, Y, S. It is the object of the remainder 

 of Klein's brilliant research to show that 

 these quantities are the simplest possible 

 elliptic time functions compatible with the 

 conditions of the problem. 



Riemann's conformal representations are 

 naturally selected as the appropriate method 

 of treatment. The first integral {t) is ap- 

 proached by mapping out v^f/on the plane 

 of complex %(,. The surface obtained is two- 

 leaved, consisting of two positive and two 

 negative distinct half sheets which cross 

 along segments of the real axis between the 

 1st and 2d root of cubic U, the 3d root and 

 infinity. 



A corresponding conformal representa- 

 tion is now made on the plane of complex 

 time, defined by the first integral above. 

 It is shown that as m moves through the 

 real axis, in the u plane, t for a single half 

 sheet of the v^ TJ surface describes a rec- 



