166 THE LAWS OP 



easy to perceive by what has been proved in the article last 

 cited, that the value of the induced density will be of the form 



r, 9 ', TX being the polar co-ordinates of the element dv, and U' (i) 

 what U" (i} becomes by changing 6" ', TX" into 0', &'. 



Still continuing to follow the methods before explained, 

 (Art. 4 and 5) we get in the present case 



and by expanding f(r'*), we have 



/(r' 2 ) = B Q + / 2 + / 4 + 



Hence, fP = B t U", and 



4-n.6-n 



2.4.6 ...... 2* -2*' 



sin 



_ 



n-2.n ...... n + 2^-4 n-l.n+1 



' 



2.4 ...... 2^' 3.5 ...... 2t'+2'+l 



Then, by giving to < all the values 1, 2, 3, &c. of which it is 

 susceptible, and taking the sum of all the resulting quantities, 

 we shall have, since in the present case V reduces itself to the 

 single term F (i) , 



'*-2 



sm I - TT 



4-n.6-n 



~4 n-l.n+1 



2.4 ...... 2t' 3 . 5 ...... 2* + 2^+1 



the sign S belonging to the unaccented letter t. 



If now V represents the function analogous to F and due 

 to the fluid on the spherical surface, we shall obtain by what 

 has been proved (Art. 3) 



v>= UK> _ 2W--- 8 ~ A ^ (0 (r>- 

 (t) representing the same function as in the article just cited. 



