Mb green, on THE LAWS OF THE EQUILIBRIUM OF FLUIDS. 7 



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

 to the sphere B whose centre is at the origin O, 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^~" 



t 



T 



in an ascending series of the powers of — . In this way we get 



^1 -« _ ^1 ^2rr {cos 9 cosff -\- sin 9 sin 9' cos (^' - -sr)] + r'^] 



l-n 

 2 



= r' " " , 



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

 after having expanded the quantity (1 — r'^f , we effect the integrations 

 relative to r, 0', and w', we shall have a result of the form 



r' = r*-'' [A-i-Br+Cf^ + Sic.] 



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

 integral relative to r' ought to be taken from r =0 to r' = r only. 



To obtain the value of F", we must expand the quantity g^-" in 

 an ascending series of the powers of — , and we shall thus have 



l-n 

 2 



g^-''={r^ — 2rr' [cos 6 cos 0' + sin 9 sin 6' cos (tst - -nr')] + r"') 



the coefficients Qo, Qi, Q2, &c. being the same as before. 



The expansion here given being substituted in P", there will arise 

 a series of the form 



of which the general term T, is 



T,= fd9'd^' sin ff QJr-'dr ^^^(l-ry.f{ry, 



the integrals being taken from r' = r to r' = l, from 0' = O to & — it, and 

 from •z«r' = to 'ar' = 27r. This will be evident by recollecting that the 



