380 Prof. H. Nagaoka on Magnetic 



whence by differentiation with respect to y, we have 



a fR'+y 2 )* 



and 



/^ CO 



\«-^J„(XR)(flt = - -•'- .. 



£ 

 1 



Consequently 



B<£ 



—J.' 



- 2 »rr 



*^o 



(a—p cos 



rftf 



o (R'+j^ 



, Xe- K yJ o (\R)vos0dXiW 



OP Jo Jo 



_ f* cos 0d<9 



These expressions can be evaluated in terms of elliptic 

 integrals, and have been first deduced by Russell*, following 

 an entirely different method, which is more direct than that 

 here given. 



The evaluation of the elliptic integrals in (7) leads easily 

 to the component Y in the axial direction and X per- 

 pendicular to it at the point a, y, o. Thus 



nW * K J i > (8) 



whe 



Y =^~-^--E + (K-E)|, 



re 



n ' 2 = (a-,.) 2 + /, 

 and the modulus of the elliptic integrals is given by 

 , 2 Aax 



The case of two coaxial circles of the same radius, at 



distance equal to the radius, is of great importance in 



obtaining uniform magnetic field, and applies to Gaugain- 



Helmholtz coils. As the field near the mean point is 



* Proc. London Phys. Soc. xx. p. 476 (1907). 



