Mutual Induction of a Circle and a Coaxial Helix, Sfc. 199 



n j-j r 2 d^fr 



" J o (l-c 2 sin 2 y>) v/(l-& 2 sin 2 ty) ' 



This expression for M@ is applicable for all values of x from to oo . 

 The elliptic integral of the third kind n is expressible by Legendre's 

 formula in terms of complete and incomplete integrals of the first 

 and second kind. Thus, if we put 



or sin/3 = 



we have (v. Cayley, " Elliptic Integrals," 183) 



These elliptic integrals, complete and.incomplete,may be conveniently 

 calculated by successive quadric transformations, as shown in 

 Cayley, Chap. XIII. 



10. Taking for the diameters of helix and circle, the values given 

 in 8, viz., 



2A' = 21-02459, 2a' = 13'01499, 



and as before 2x 5*02480, 



Professor Ayrton and I made the calculation of Me by equation (6) 

 with the following results : 



c = 0-9719222, 



~k = 0-9615024, 



V = 0-2747959, 



L sin (3 = 9-9326156, 



F (&) = 2-7109750, 



E (A;) = 1-0840174, 



F (&',) = 1-0393881, 



E (&', /3) = 1-0168643, 



fc' 2 sin 8 cos 8 



L (F 0) = -0-5051433, 

 c 



c /2 



-y (F n) = 0-8616240, 



?JrJ? = 1-7598494, 

 M = 22913 '59. 



