of a Current running in a Cylindrical Coil. 209 



functions of the first and second kinds. Thus, for a 

 complete function with the parameter — m it is known 

 that, if we put m=—k /2 sin 2 e, 



U(-m,k)*mK+ sr J± {5 +KK' e -KE'«-EK'A, (12) 



v rsinecose L2 J 



where A' e stands for Vl— <&'" 2 sin 2 e, and K e , E e stand 

 for incomplete functions of the first and second kinds with 

 modulus k' and amplitude e. 

 In the present case, 



m = cos 2 0' k ! = - = -: — tt ; 

 p smu 



.*. e = j and 



II(-cos» ff, k)=K + ^ ™Z oos e {I T KK;- KBi-BKi}.; (13) 



so that (7) becomes 



^ = ^{KEi + EKi-KKi-Tr} + P E+p' cos cos 0'. K. (U) 



It is evident that we may define the position of any point, 

 P, in the plane of the figure by means of the two coordinates 

 k and 0. Thus we have 



2a 

 9 ~~ A o + k'cos0' 



Hence 



cos0' = A e , 



z = k'p sin 0, (j ! = k'p. 



~ = -Ti—£-* -7r{E + ^sin(9(KE' 4-EK'.-KK^-7r) 

 4m A' d + /dcos0 l v e - e a j 



H-^cos6>.KA' }. . (15) 



This, then, is the expression for the potential at any point 

 in terms of the coordinates (£, 0) of the point. In par- 

 ticular, it gives the value (9) for any point on the per- 

 pendicular through B to the plate, since for such a point 



0=~, and then the coefficient of A;' sin within the brackets 



7T 



is equal to —&, hy Legendre's well-known relation between 

 the complete complementary functions, viz., 



KE' + EK'-KK'=£, 



2' 



whatever the modulus k may be. 



