356 PROCEEDINGS OF THE AMERICAN ACADEMY 



This same method of approximation may be applied to equation 

 [XV.]: 



2 r- ^ ^ a cos a ' 



. , a' + i 1 dPi(a) p p -, 



I i + 1 r* + ^ d cos a ^ ^ "- -^ 



Substituting q for those terms depending upon x and y, 



Q' = 2^^, ^P, (^) + &c. + 2,r?,-^,P.(^), [XXL B.] 



the value for one turn. Let ff be the mean value for all the turns 

 within the coil of cross-section 1 7, as deduced from the expression 



Then, for the potential due to the whole coil, 



1 ^ .-. , „ , ^ 1 



O 



' = Srry, ^, P, (6) + &c. + 2nff, ^, P, (6) ; 



or, calling y/ =z ^ng^^., 



Q' = i7/ ,-, P. {&) + y/ i ^. (^) + &c. ; [XXIL] 



sin^a ^dP.{a) 1 »/ o ,0 

 ^1 2 (/cos a 2 2^ ^-^ 



5'i' = -/+TV'^r; [XXIIL] 



^r; = 3;./(x^ - If) +1^(2x2 - 3/) + %rihf. 



Equations [XIX.] and [XXIL] give the value of the potential at 

 any point due to the current in a coil of any shape. 



