of a Current running in a Cylindrical Coil, 207 



so that 



D 



dcj> 



I = -27TZ+ ( V* 2 + I>#+ {Ct 2 -^) \ 



™ Jo Jo 



+ l *-^irfhr * (3) 



Again, if we put <£ = 2ft>, we have 



D = (a + ocf— 4a# sin 2 <w ; 



and if the distances PA, PB are denoted by p, pf, respectively, 

 we have 



\,ax = p 2 —p 12 . 

 so that 



z* + ~D = pt-Qfi-p 1 *) sin 2 ft>. 

 Let 



#=!-£!; &' 2 = ^; Aft)= Vl-^ 2 sin 2 ft). . (4) 



9 5 '" « 2 



Then (3) becomes 



p 2 ™ +2 , E+ V-w^ g-;. r- J a . £, (5) 



r r ^o \— -sm^fi) 



where E and K are the complete elliptic integrals of the 

 second and first kinds with modulus k. 



The integral in (5) is the complete elliptic integral of the 



(a + a) ' 

 parameter is numerically greater than the modulus ; and we 

 shall find it convenient to convert the integral into one 

 in which the parameter is less than the modulus by the 

 well-known rule that a function with parameter n can be 



converted into one with parameter — . If the angles PBA 



n 



and PAB are denoted by and 6', respectively, we see that 



third kind with modulus k and parameter — fn _^_ , 2 . This 



n 9 the parameter in (5), is — ■ — q-m ■> so that the new 



COS' 



6" 



parameter will be simply —cos 2 6', 

 Now we have the general result that 



BM + n^.)-^i»-(i%L-) + K Wl 



