172 THE LAWS OF 



Application of the general Methods to circular conducting 

 Planes, &c. 



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 immediately 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 



/ being the characteristic of a rational and entire function of the 

 degree s ; #', y the rectangular co-ordinates of any element da 

 of the plane's surface, and /, & the corresponding polar co- 

 ordinates. 



Then we shall readily obtain the formula 



n 1 



a > 



where r t 6 are the polar co-ordinates of p, and the integrals are 

 to be taken from 0' = to ff = 2-Tr, and from r to r = 1 ; 

 the radius of the circular plane being for greater simplicity con- 

 sidered as the unit of distance. 



Since the function f(x, y'} is rational and entire of the de- 

 gree Sj we may always reduce it to the form 



/(', y) = A M + A (l) cos & + A cos 2ff + A cos 3(9' + 



+ J3 (1) sin & -f B (z} sin 20' + B (3) sin BO' + (24), 



the coefficients A w , A (l \ A ( *\ &c. 75 (1) , J5 (2) , .Z? (3) , &c. being func- 

 tions of r only of a degree not exceeding 5, and such that 



'*+ <V 4 + &c.; A (l) = ( (1) + <V 3 + 2 (1) r' 4 +) / ; 

 + & a ( V 4 + &c.)r'; B w = (5 (2) +5 l ( V 8 + &c.) r' 2 . 



We will now consider more particularly the part of V due to 

 any of the terms in / as A (i) cos iff for example. The value of 

 this part will evidently be 



