w 



184 Mr CHALLIS's RESEARCHES IN THE THEORY 



considerations with which we commenced this investigation. Neglecting 



powers of I above the first, this quantity becomes V \\ ^ J . 



Consequently 



a 



But S = SZcospPs = SZ X , 'g.J p' Hence 



w'-w _ „ / \-d 1 ¥-a^ 1 



~IZ U +«' + *'■ / + r "^ l+«2 + fr^'r 



If now a, ft, 7, be the angles which the axis AZ makes with 

 Ox, Oy, 0%, respectively, we have 



Hence passing from differences to differentials, 



-7- = (COS^'V — COS'o)^ + (cOS^/3-COS*a) - (1). 



d% ' ' l + r ' r ' 



So if d, /3', 7', be the corresponding angles for the axis of Y, and 

 a", /3", 7", for that of ^, v the velocity in F", and u that in ^, we shall 

 have by a like process, 



^ = (COSV - COs'a') ,— + (coS^/3' - COS^a') ....... (2) , 



^ = (cos^y- COs'a") y^ + (cos'/3"-COS^a") ^ (3). 



