470 



SCIENCE. 



[N. S. Vol. VII. No. 171. 



unique method of presentation adopted — 

 all the curves being worked out in form of 

 stereoscopic diagrams — -endows his results 

 with an objective reality ; and when one 

 remembers that these complex carves reach 

 only especially simple cases of gyroscope 

 motion, one may get some notion of the 

 difficulty of the problem involved. 



Turning now, from Greeuhill's necessarily 

 cumbersome equations for the approachable 

 part of the problem of rotation, to Klein's 

 little book, one is astonished in finding the 

 most general aspects of the subject treated 

 almost without computation and in so 

 little space. This astonishment, however, 

 Is in a manner relieved on learning that the 

 discussion remains formal throughout, that 

 much of it is epitomized, many proofs 

 sketched in, and that the reader is sup- 

 posed to be thoroughly versed not only in 

 dynamics, but familiarly conversant with 

 the theory of complex variables, with 

 elliptic integrals and functions particularly 

 in reference to their derivation from f} and 

 <T-functions, their generalization in terms of 

 automorphic functions, and to be as well 

 read as possible in the geometry of hyper- 

 space. The reviewer, who makes no special 

 pretense to these accomplishments, has 

 taken up Klein's remarkable book, since it 

 professedly appeals to physicists and has 

 groaned through it. He ought, therefore, 

 at the outset to confess to a feeling of hos- 

 tility because of its unbending mathemat- 

 ical aloofness. In a book with a professed 

 missionary purpose it is not unreasonable 

 to expect just a little condescension in favor 

 of the kind of mathematics with which 

 physicists are, as a rule, more familiar. 

 Judicious annotation either on the part of 

 Professor Klein himself or by Professor Fine 

 would have speeded the propagandist. I 

 doubt whether everybody will ' at once ' 

 recognize the elliptic integrals of pages 28, 

 29 as being normals of the third type, par- 

 ticularly when the notation of Legendre 



and Jacobi is different. It would have cost 

 but little to give the expanded form of the 

 ff -function. If the reviewer is not incorrect, 

 Weierstrass's original notation was in terms 

 of Abel ian functions. The tremendous de- 

 velopment of elliptic functions is out of 

 proportion with their application to natural 

 phenomena. Meeting them rarely one for- 

 gets them. Memory peters out like the 

 infinite series of a '^-function. Mathemati- 

 cians will do well to observe that a reason- 

 able acquaintance with theoretical physics 

 in its present stage of development, to men- 

 tion only such broad subjects as electricity, 

 elastics, hydrodynamics, etc., is as much 

 as most of us can keep permanently assimi- 

 lated. It should also be remembered that 

 the step from the formal elegance of theory 

 to the brute arithmetic of the special case 

 is always humiliating, and that this labor 

 usually falls to the lot of the physicist. 



To return from this paroxysm to the 

 splendid research under discussion, let us 

 note first that Kleiu begins his analysis 

 with the top spinning on a sharp friction- 

 less pivot, so that a simple point in the 

 axis (not the center of mass) is fixed. To 

 this special case the first three lectures are 

 devoted. In the fourth the restriction is 

 cut loose. Klein's method is to consist in 

 a far-sighted choice of coordinates, and the 

 first lecture is, therefore, a comparison of 

 available systems with their mutual trans- 

 formations. The Cartesian definition of 

 three movable in terms of three fixed coor- 

 dinates with a common oi-igin in the fixed 

 point, by the 9 direction cosines considered 

 as functions of time, is first taken up. 

 The corresponding transformation scheme 

 is thereafter expressed in terms of Euler's 

 »?, <p, 4', parameters ; in terms of the rota- 

 tional or quaternion parameters, and fi- 

 nally in terms of Klein's new parameters, 

 which are introduced as follows : x, y, z, 

 and X, Y, Z, being the coordinates of given 

 points on a fixed and a movable sphere, re- 



