12 Mr green, ON THE LAWS OF THE EQUILIBRIUM OF FLUIDS. 



retains the same numerical value, and merely changes its sign when 

 we pass from the element dv to a point diametrically opposite, where 

 the co-ordinates x, y , % are replaced by - x , - y' , —% \ it is easy to 

 see that the function V-^, depending vxpon /s,, possesses a similar property, 

 and merely changes its sign when x, y, %, the co-ordinates of p, are 

 changed into - x, —y, — as. Hence the nature of the function Vi is 

 such that it can contain none but the odd powers of r, when we sub- 

 stitute for the rectangular co-ordinates x, y, %, their values in the polar 

 co-ordinates r, 6, ■zs. 



Having premised these remarks, let us now suppose Vx is divided 

 into two parts, one V^ due to the sphere B which passes through the 

 particle p, and the other V" due to the exterior shell aS*. Then it is 

 evident by proceeding, as in the case where p = (1 - r"^Yf{i% that Vi 

 will be of the form 



the coefficients A, B, C, &;c. being quantities independent of the variable r. 



In like manner we have also 



F/' = fr'^dr'ae'dsr' sin ff {\-ry .r'>\,{r'')g^-''; 



the integrals being taken from r' = r to r = l, from 6' = to 0' = 7r, and 

 from Gr' = to 'z<r' = 2 7r. 



By substituting now the second expansion of g^" before used (Art. 1.), 

 the last expression will become 



r," = t; + Ti + r. + ^3 + &c. 



of which series the general term is 



T, = fd9'dw' sin ff Q, fr"-dr' (1 - ry ^ x/. {r"). 



Moreover, the general term of the function \l^ {r'-) being represented by 

 Air'^\ the portion of 1\ due to this term, will be 



(a) r fdffdw' smO' Q,Atjr''-''^''-Ulr' {l-ry-, 



•the limits of the integrals being the same as before. 



