276 PROFESSOE A. G. GREENHILL ON THE 



37. The simplest cases may be cited before proceeding to the general theory. 



,1=4, v = K + K'i (1), 



(I - 



_ , / - * - Bin -i 



V 2(K-<) 



and in the associated dynamical applications 



M P = X /(K - x"\ x = KSD. (K - mt) (3), 



/c 



M k e (vt -" = x/i (1 - .x) ( + .x) + i V\ (1 + ^l/T^T) (4), 



reducing to HALPIIEN'S algebraical herpolhode when the secular term pt is cancelled 



by putting p = 0. 



In the associated motion of the top (' Annals of Mathematics,' vol. 5) 



m 



(5). 



But the quadric transformation 



(1 K)X r/ , 2 X/ 



' - en |(1 + K)i>it,y\, y = --V 



makes 



thus effecting the identification with the result, p. 147, ' (Euvres/ 2. when the sign of 

 ABEL'S K is changed so as to obtain the circular form. 

 For the division-values 



j-^ cn|K'= Vr -^ f , dniK's=--/K (8); 



iK' = Z (liK' = 1(1 -- /c), zsiK'= -zcK' = i(l + K) (9); 



iK'_(l+ w )t HiK'_(l-K) HiK'_ _K* 

 00 2V 2^ HK' 2*(H-)* 



