THE EQUILIBRIUM OF FLUIDS. 169 



and the corresponding value of p by what precedes will be 



> 

 2sm 



7T 



x 



+ &c. tn tn/.i; 



ZT (0) , U' (l \ U' m , &c. being what J7" (0 >, t^ 1 *, Z7" w , &c. become 

 by changing 0", or" into 0', -cr', the polar co-ordinates of the 

 element dv. But, since we have generally 



sn 



(Mec. Gel. Liv. in.) the preceding expression becomes 



sm[_-7r] 



/~2 i\ 2 



the integrals being taken trom 0"=0 to 0"=7r, and from -sr" to 

 if n 



OT = 2?T. 



In order to find the value of the finite integral entering into 

 the preceding formula, let R represent the distance between the 



two elements d<r, dv; then by expanding -g in an ascending 



r 



series of the powers of - we shall obtain 



'a 2 - 2ar {cos 0' cos 0" + sin & sin 0"cos (r f - r") j + 1" 



) r/ * 

 o V a i > 



Liv. in.). Hence we immediately deduce 



If now we substitute this in the value of p before given, and 



7 2 '2 



afterwards write -, and ^ 3 in the place of their equivalents, 



