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



quantity condensed upon it, that its density may every where be repre- 

 sented by 



ftl 2 \ „-4 „_2 4-n 



Application of the general Methods to circular conducting Planes, &f:. 



10. Methods in every way similar to those which have been used 

 for a sphere, are equally applicable to a circular plane as we shall im- 

 mediately proceed to show, by endeavouring in the first place to determine 

 the value of V when the density of the fluid on such a plane is of 



the form 



p = {\-ry.f{x',y'): 



f being the characteristic of a rational and entire function of the degree * ; 

 x\ y' the rectangular co-ordinates of any element dcr of the plane's 

 surface, and r', & the corresponding polar co-ordinates. 



Then we shall readily obtain the formula 



r= ff^ = r rrdr'd9'{l-ry.f{x',y') ^ . 

 '' g"'' ''■^ {f^-Zrr' cos {9-9') + r"f^' 



where r, 9 are the polar co-ordinates of p, and the integrals are to be 

 taken from 9' = to 0' = 27r, and from r' = to /•' = !; the radius of 

 the circular plane being for greater simplicity considered as the unit 

 of distance. 



Since the function /{x', y') is rational and entire of the degree j, 

 we may always reduce it to the form 



(24) f{x', y') = A^°^ + A^'^ cos 9' + A^'^ cos 20' + ^*'' cos 39' + 



+ ^« sin 9' + B'-'^ sin 29' + B^'^ sin 30' + 

 the coefficients A'-''\ A^'\ A^'\ &c. B^'\ B^% B^\ &c. being functions 

 of r' only of a degree not exceeding *, and such that 



^('•'=«'o°' + «<V^ + 4"V'* + &c.; ^« = («?> + alV + 4'V'V)/; 

 ^(» = ( jw -I- J(/)r'^ + i(»r'^ + &c.) r' ; B'^ = {bf> + hfr" + &c.) r'\ 



