THE EQUILIBRIUM OF FLUIDS. 125 



the integrals being taken from OT' = to -57' = 27r, from & = 

 to 6' TT, and from / = to / = 1. 



Now V may be considered as composed of two parts, one 

 V due to the sphere B whose centre is at the origin 0, and 

 surface passes through the point p, and another V" due to the 

 shell S exterior to B. In order to obtain the first part, we must 

 expand the quantity g l ~" in an ascending series of the powers 



of . In this way we get 



l-n 

 gl-n = |yj _ 2rr > J cog Q CQS fl' + Q ' m Q gm Q> CQS ^ -)}+ r' 2 ] 2 



If then we substitute this series for g 1 ^ in the value of F', 

 and after having expanded the quantity (1 T*' 2 )* 3 , we effect the 

 integrations relative to /, 0', and -or ', we shall have a result of 

 the form 



V = r*-" {A + Br z + W + &c.} 



seeing that in obtaining the part of V before represented by V, 

 the integral relative to r ought to be taken from r' = to r =r 

 only. 



To obtain the value of F", we must expand the quantity 

 ^ 1 ~ n in an ascending series of the powers of , and we shall thus 

 have 



l-n 



g l ~ n = (r* 2rX [cos 6 cos & + sin 6 sin 6' cos (r ')] + r' 2 ) * 



the coefficients $ , ^, $ 2 , &c. being the same as before. 



The expansion here given being substituted in F", there will 

 arise a series of the form 



of which the general term T 8 is 



T, =Jd0' thi sin ff QJr" dr' (1 - 



