DETERMINATION OF M BY ELLIPTIC INTEGRALS. 127 



the substitution of these values in Neumann's formula will give 



aa' cos < - 



r** /- 



as, I 



Jo Jo 



- 2 a cos < - 

 This integral is given by the expression 



(19) 



in which 



and F and E denote complete elliptic integrals of the first and second 

 kind with the modulus k. 



If r-i and r 2 are the extreme values of r, we have 



r\ = (a + aj + x\ r\ = (a- aj + x\ 

 If we put 



cos y = , or k = sin y, 

 r i 



formula (19) may be written 



(20) y[. = A.TT\J aa' ( sin y : IF + -: E 



L\ smy/ smy J 



Taking k = , , which gives k = , we have, in the case of 



the complete integrals, the ratios 



The substitution of these values in equation (19) gives, for the value 

 of M, an expression which is sometimes more advantageous : 



(21) M = 



Writing equation (20) in the form 

 (22) _J 



^TTsaa sm y 



we see that the second expression is simply a function of the angle y. 

 The following tables, taken from the second edition of Maxwell's 

 Treatise, give logarithms of the values of the first number for values of 

 y, varying from 6' to 6', from 60 to 90. 



