OP ARTS AND SCIENCES. 353 



Substituting the value of V and V from equations [XII.] and 

 [XIII.] for V in the expression 



ld(rV) 



Q — — J } 



a a r 



and differentiating, we find 



o = -2,{i-cos„ + *c, + ^:^=i'L^p.w}, [XIV.] 



Q' = 27r sin^a \ ^ -,Pi (6) , ^^ ' + «Sjc. 

 \^2 r^ 1 V ^ J cos a ' 



+ 1 ^^ZZlMp,.(^)|. [;xv.] 



I l-\-\ ?•» + ! rf COS a » ^ >' j L J 



These are the most general expressions for the potential at any point 

 whatever, due to a single coil of wire. Equation [XIV.] must be 

 employed if r is less than a, or if the point lies within the sphere of 

 which a is the radius and [XV.] for all points outside. 



If the origin had been taken at the centre of the circular coil, a 

 would have been equal to f or 90^. Substituting this value for a in 

 [XIV.] and [XV.], 



c {. nAo , r , sin2 90° r' f/P,(90°) „ ,,, ") 



Q = — 27r ^ 1 — COS 90° + &c. H '. ;. ■ '\,,o Pi (6) \ ; 



( ' ' I a' (/cos 90 '^ '^ J 



C = 2;r sin2 90° \\-,P, {6) '^^-"-^2 + &«• 

 ( 2 r^ ^ ^ ^ a cos 90 ' 



T~i_|_l r'+' rfcos90° '^^ j ■ 



d P 



Noticinsr that cos 90° =: 0, and that all the even orders of -; 



° c/ cos a 



are multiplied by cos a, and therefore equal to 0, we have 



= -2^|l+^P,(^)-^^>3(^) 





8 a5 « ^ '' IG a'' 



^' = 2-{2 rJ^^(^)-|?^^3W 



+ ^'^Ps(^)-^'^^r(^) -^&c. 



VOL. XXI. (n. s. XIII.) 23 



