April 8, 1898.] 



SCIENCE. 



473 



the t plane, in order to show the limits 

 traced by the pole point on the C sphere. 

 Indeed, the investigation is advantageously 

 -thrust back a step further by considering C 

 as the image of the corresponding four half 

 sheets of the Vf surface. It is hardly 

 possible to follow Klein through this in- 

 volved discussion here without reproducing 

 his figures and computation in full. Sufl&ce 

 it to say that the stereographic projection 

 of the C image from the top (s = r, C = oo ) 

 of the z axis, on the xy plane, is mapped 

 out in correspondence with the parallel- 

 ogram of periods on the plane of complex 

 time, or for each point of the two positive 

 a,Dd negative half sheets of the ^U sur- 

 face. 



The lecture concludes with a demonstra- 

 tion showing that a free body in hyperbolic 

 non Euclidian space may be so fashioned as 

 in real time to carry out the actual motions 

 of the top. The form of such a body and 

 the forces which actuate it are specified. 

 Klein lays great stress on the beauty of this 

 generalization. 



In the fourth lecture, as already inti- 

 mated, the top is set spinning on a hori- 

 zontal plane with its point of support free 

 to roam at pleasure, so that the top now 

 has 5 degrees of freedom. In any case, 

 however, the horizontal motion of the 

 center of mass is uniform, and this point 

 may, therefore, without essential restriction 

 be considered fixed. But if the origin of 

 coordinates be taken at the center of mass 

 the problem returns to 3 degrees of freedom, 

 with the difference that a new term equiva- 

 lent to its vertical motion must make its 

 appearance in the expression for kinetic 

 energy. Hence a new treatment of the 

 equations of motion is necessary, and if 

 Eulerian coordinates be again introduced 

 the method sketched in the 2d lecture is 

 applicable throughout. The result for t, 

 <f, 4> now, however, lead to hyperelliptic in- 

 tegrals, as for instance, 



-/ 



^/ (J + Ps) — Psu' 



du 



(where s is the distance between the cen- 

 ters of support and of mass and P the 

 static moment), with a corresponding 

 increase of the difficulty of the prob- 

 lem. The two new roots in the in- 

 tegrand thus make the corresponding 

 Eiemann surface two-leaved with six-branch 

 points ; but Klein shows that the param- 

 eters a, /?, /, 3 are again singularly adapted 

 for the treatment of the present case, with 

 this fatal difference, that for a single point 

 in the t plane there correspond an infinite 

 number of value of u. Hence as u is no 

 longer a single valued function of t, it be- 

 comes necessary to seek a new function of 

 which complex t, a, /9, r, S shall all be single 

 valued dependents. Such functions are the 

 automorphic functions (ij) obtained from 

 elliptic functions by generalizing their perio- 

 dicity . The line of argument above can now 

 be broadened ; construct in the ij plane a 

 rectangular hexagon which is the image of 

 a half-sheet of the Eiemann surface on the 

 u plane, and which on reproduction covers 

 the plane of complex tj conformally and 

 simply. Then to each point on the rj plane 

 there corresponds a single point on the 

 Eiemann surface ; or 



u, s/TT, v/ 1 -f Ps — Psu\ a, /?, r, ^, 

 are all single valued functions of rj. Thus 

 ri quite replaces the t in the special case, 

 and Klein carries out his analogies in de- 

 tail by expressing the automorphic functions 

 in terms of quotients of what he calls 

 prime forms. Hence a, /J, y, d are now 

 given in terms of quotient of simple prime 

 forms of 5j-fanctions, while they were above 

 given as quotients of simple <T-functions. 

 The fall geometry of the case is not carried 

 out in these lectures, however, and Klein 

 regrets that the development of the auto- 

 morphic functions has recently fallen into 

 abeyance. 



