Mr green, on THE LAWS OF THE EQUILIBRIUM OF FLUIDS. 9 



it is clear that none of the terms last mentioned can enter into V, 

 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 



m —^ ^-^ Air'.fde'd-Br' smd'Qs. 



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, are of the form 



r(#+2-y-|) r(/3 + i) 



(4) ^ Atr'^fde'dTir' sin ff Q,r, 



2r(^ + /3 + 3-*'-|) 



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

 s, and as Qu- is always a rational and entire function of cos 9', sin 0' 

 cos w', and sin 9' sin -sr', the remaining integrations may immediately be 

 effected. 



Having thus the part of T'a,- due to any term Atr'''* of the function 

 y(r'*) we have immediately the value of T'.^' and consequently of F'", 

 since 



r"= u'+ t:+ t:+ t:+ T:+kc. -, 



U' representing the sum of all the terms in F" which have been rejected 

 on account of their form, and T,' T,' T,' the value of T, Ty T„ &c. 

 obtained by employing the truncated formula (2) in the place of the 

 complete one (2). 



But -v=v'+ V" = r'+ u^Ti^ r;+ t: + 7v+ &c. 



or by transposition, 



r-T:-T;-Ti-Ti-hc.=r-YU, 



and as in this equation, the function on the left side contains none 

 but the even powers of the indeterminate quantity r, whilst that on 

 Vol. V. Part I. B 



