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SCIENCE. 



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



tangle in the t plane, whose position and 

 sides (periods) are determinate when the 

 time integral is made definite. Four ad- 

 jacent and congruent rectangles in the t 

 plane correspond to the four half sheets of 

 the Eiemann surface. Finally for any 

 march of u around the segments between 

 successive roots of U„i receives a constant 

 increment, such that the complete image in 

 the t plane covers the whole infinite t sur- 

 face with congruent adjacent rectangles 

 which nowhere overlap. Hence the im- 

 portant conclusion is accentuated that 

 whereas for each point u there correspond 

 an infinite number of values of time (<), 

 for each value oit there corresponds but 

 one value of u, and hence u like >/Tj are 

 single-valued, doubly-periodic elliptic time 

 functions. 



Klein next takes up the relations of <p 

 and <p to t, a, problem much more complex 

 but one in which he scores his most signal 

 triumph. Introducing his own parameters 

 a, /?, Y, (5, already defined in terms of Euler's 

 coordinates, Klein obtains normal integrals 

 of the third type without further reduction, 

 while the four logarithmic discontinuities 

 are assignable, one each to log «, log /S, log;', 

 log 5, with a common logarithmic discontinu- 

 ity at u = 00. The transformation thence to 

 exponentials (a, /?, y, S) is equivalent in 

 Klein's interpretation to a passage from 

 elliptic integrals to elliptic functions, and 

 now he is able to avail himself of the quo- 

 tient of two ff-functions (each of which 

 contains null-points only), together with an 

 exponential time factor to fully express his 

 parameters. They severally vanish for 

 M = ± 1 and became oo for t= 0, one in 

 each parallelogram of periods. Finally 

 the 9 direction cosines known in terms of 

 a, /?, y, S are, therefore, also expressed in 

 term of quotients of ff-functions. 



Having thoroughly unveiled the charac- 

 ter of his parameters «, /9, y, S, Klein pro- 

 ceeds with their application. The Z pole 



of the moving sphere is preferably selected 

 for tracing top curves. At this point 

 Z = 00, and, therefore, the paths on the 

 fixed sphere become C = a/y. Hence ? too 

 is at once expressible as a single quotient 

 of single valued o--functions, together with 

 an exponential time factor. An essential 

 simplification has thus been achieved over 

 all preceding methods. Hermite in his 

 treatment of the stereographic projection 

 of the Z pole needed functions as complex 

 as products of Klein's functions, while even 

 in the hands of Jacobi the first degree of 

 complexity reached only the specialized 

 case ofaPoinsot motion, i. e., rotation rela- 

 tive to a fixed center of mass. 



A point of cardinal interest in this lecture 

 is the investigation of the rolling and the 

 fixed cones (polhode and herpolode of the 

 top motion), which, by Poinsot's theorem 

 are adapted to describe all rotations about 

 a fixed center. The object in quest here is 

 an expression of the rotation about the in- 

 stantaneous axis, or preferably of the com- 

 ponent rotations about the three movable 

 axes, X, Y, Z, fixedin the body in terms of 

 Klein's parameters «, i^, y, S • i. e., virtu- 

 ally to refer the rotation to the axes x, y, z, 

 fixed in space. The results again show the 

 remarkable adaptation of the new param- 

 eters to the problem in hand. When the 

 three principal moments of inertia are equal, 

 both polhode and herpolode turn out to be 

 elliptic plane curves of the first degree. 

 Thus both polhode and herpolode of the 

 top's motion would be polhodes of two cor- 

 responding Poinsot motions ; recalling the 

 theorem of Jacobi that the motion of a top 

 may be expressed as the relative divergence 

 of two Poinsot motions. 



Finally the motion of the polepoint, al- 

 ready briefly sketched for motion in real 

 time, is resumed, in relation to complex 

 time, to fully bring out the power of the 

 elliptic functions «, /?, y, '5. Attention is 

 first given to the parallelogram of periods in 



