232234] ELECTROMAGNETISM. 457 



and writing 



K Z 



we have finally 



& -E(K) 



K 



where E and F are the elliptic integrals 



T2 



= 1 V 1 tf 2 sin 2 -v/rcfyr, -^W = 



Jo 



Jo Vl /e 2 sin 2 i/r 



These definite integrals are functions only of the parameter K, 

 and their values have been tabulated by Legendre for various 

 values of K. If we put 



<# + (& 



r 2 and r 2 are the maximum and minimum distances of points 

 on the circumferences of the two circles from each other. The 



expression M/^TT *jRJRq being a function only of K and therefore 

 of 7, has been tabulated by Maxwell as a function of 7. (Treatise, 

 Vol. 2, Art. 701.) 



We may also find the value of M in a series of zonal spherical 

 harmonics by means of the series of 215 by differentiation with 

 respect to r and integration over a spherical segment bounded by 

 the second circle. For a full treatment of the properties of 

 circular coils the reader is referred to Maxwell's Treatise, to 

 Mascart and Joubert, Lessons on Electricity and Magnetism, and 

 to Gray, Absolute Measurements in Electricity and Magnetism, 

 where a great variety of formulae will be found. 



234. Non-linear Currents in Parallel Cylinders. If 



the expressions in the two preceding sections be used to find 

 the self-inductance of a linear circuit we find a difficulty, for on 

 putting a = in 232, the expression becomes logarithmically 

 infinite, while on putting a 0, R^ = J? 2 in 233, K becomes unity, 

 the elliptic integrals reduce to trigonometric, and F (K) becomes 

 logarithmically infinite (log tan w/2). This is easily seen to be 

 the case for any linear circuit, for if dsi and ds z traverse the 

 same circuit there is an infinite element in the integrand, and, 



