246 PROFESSOR A. G. GREENHILL ON THE 



Thus- for /u, = 3 in 6 (5), putting u z + \a, and a = 36, 



= cos- 



A=-6, B = 36 2 , 2C=-4& 3 -_p, D = 6(4& 3 +p) (7). 



For instance, fr = makes fj. = 0, H = 0, /"j = 0, and the orbit is 



r cos | = c* (8), 



described under a constant central force. 

 For fj. = 5 in (7), 7, putting 



w = 5z 1 2x (9), 



= cos- - 



~5~ v/ ' ! ~ '" 3 + (/ "" ' r) ?y2 + 8 (~ 1 + H* + x 2 ) w 



+ 4(4 + 3x)(-l + lla; + a5 2 )} (10), 



and to make /JL = 0, put x = 3, so that with w = 5w, 



2 _,-- M 



, cos 



o b 



2 _,-- M + 4 / / , x 

 , cos - v/ (M + 2) 



= - sin l - Y/ ( ?/, 3 -|- 2'~ 8u -{- 4) (H)> 



an orbit described by a body attached to an elastic thread, which is led through a 

 fixed origin, which can be written 



Gr* cos J 6 (4r~ or + a-) N / (2?" + a) (12), 



GH sin f = (/ + a) ^/ (4r 3 8r e + 2a~v ft 3 ) (13). 



So also for /M, = 7. 



THE SECOND STACJE. 



22. But in the dynamical applications, such as POINSOT'S herpolhode and JACOBI'S 

 associated motion of the top, the integral of the Second Stage is required, corre- 

 sponding to an even value of p., and S can now be resolved into its factors 



a A / . \ /.. * \ ( v 9 \ /ll 



O 4 ^,S S l ) ^6 f> 2 ) (f> S<i) (L), 



while in most cases of the applications 



s 1 ><r>s i >s> s s (2), 



