M THE EQUILIBRIUM OF FLUIDS. 131 



If now we effect the integrations relative to r by means of 

 the formula (3), Art. 1, and reject as before those powers of the 

 variable r, in which it is affected, with the exponent n, since 

 these ought not to enter into the function F I} the last formula 

 will become 



^-V(+i) 



r 8 fd0'dv r sin0'Q 8 A t (a'), 



and as T^ ought to contain none but the odd powers of r, we may 

 make s = 2s -f 1, and disregard all those terms in which 5 is an 

 even number, since they will necessarily vanish after all the 

 operations have been effected. Thus the only remaining terms 

 will be of the form 



r /4-. + g .-.*rx ; . ,::,-,,,; 



2. 



2 } 



where, as A t and Q zs , +l are both rational and entire functions of 

 cos & , sin & cos -sr', sin & sin-cr', the remaining integrations from 

 0' = to & = TT, and CT' = to *r' = 2?r, may easily be effected in 

 the ordinary way. 



If now we follow the process employed in the preceding 

 article, and suppose Zy, T 7 /, T^ &c. are what T , T I} T z , &c. 

 become when we use the truncated formula (a) instead of the 

 complete one (a), we shall readily get 



In like manner, from the value of V % before given, we get 

 F 2 " ^pWdffd*' sin ff (1 - r"*Y $ (r' 2 )/-" ; 



the integrals being taken from r =r to r=l, from ^' = to 



0' = TT, and from OT = to OT = 2?r. 



Expanding now g l ~ n as before, we have 



92 



