THE EQUILIBRIUM OF FLUIDS. 127 



which the quantity r is affected with the exponent n. But, as 

 in the case under consideration, n may represent any number 

 whatever, fractionary or irrational, it is clear that none of the 

 terms last mentioned can enter into F, seeing that it ought to 

 contain the even powers of r only, thence the terms of this kind 

 entering into V" must necessarily be destroyed by corresponding 

 ones in V. By rejecting them, therefore, the formula (2) will 

 become 



s - 



(2'). 



But as V ought to contain the even powers of r only, those 

 terms in which the exponent s is an odd number, will vanish 

 of themselves after all the integrations have been effected, and 

 consequently the only terms which can appear in V t are of the 

 form 



(4); 



where, since s is an even number, we have written 2s' in the 

 place of 5, and as Q& is always a rational and entire function of 

 cos 0', sin & cos /, and sin ff sin OT', the remaining integrations 

 may immediately be effected. 



Having thus the part of T'& due to any term A t r tat of the 

 function f(r*) we have immediately the value of T' - and conse- 

 quently of V" 9 since 



V"=U'+T 9 '+TJ+ 2y + T e f + &c.; 



V representing the sum of all the terms in V" which have been 

 rejected on account of their form, and T ' T^ T the value of 

 T TI T^ &c. obtained by employing the truncated formula (2) 

 in the place of the complete one (2). 



But - F= V'+ V" =V' + U'+ T '+ Zy + TJ+ T^ 

 or by transposition, 



7_ zy _ Tj - Tl - T; - &e. = V + U'; 





