208 Prof. G. M. Minchin on the Magnetic Field 



where e is the amplitude of each of the two functions of the 



third kind (denoted by II), and «= (1 +n)l 1 + — )• Hence 



for complete functions ( e=^ J we have 



n(»)+nf-) = -^ ? -+K. 



V cos 2 &) \a + xj p 2 



But here 

 so that 



nf — k *)"p-?±£+K-n(-co$*ff), . . (6) 



\ cos 2 0'/ 2 z a—x 

 and (5) becomes 



I = _« + 2pE + 2^.K-2£^n(-c«f»*',A). (7) 



m r a -f # p a-\-x 



This expression holds without ambiguity for all positions of 

 the point P, and it shows that for all points 



y 



on the axis, OV, of the plate, . - = 2ir(p-z), . . (8) 



lib 



y 



on the perpendicular through B, - = 2pE — ttz, . . (9) 



lib 



Y 



in the plane, between and B, . -=2pE + 2p'K, . (10) 



lib 



Y 



in the plane, beyond B, ... — = 2pE — 2p'K ; . (11) 



so that the points, occupying any of these positions, at which 

 Y has any assigned value can be easily found. Thus, to find 



the point on OV at which — has the value C, we have for 



• nx 



this point 



C 



Q 



Hence draw below AB, parallel to it and at the distance ■%- , 



LIT 



a right line, meeting VO produced in 0' ; then the perpen- 

 dicular to AO' at its middle point meets OV in the required 

 point. 



Now every complete elliptic integral of the third kind 

 can be expressed in terms of complete and incomplete 



