442 Mr. A. Russell on the Magnetic 



case, therefore, o£ N x infinitely great, that is, when the helix 

 is practically identical with a cylindrical current sheet of the 

 same axial length and diameter, we see that the formula for 

 two cylindrical current sheets must also be mNxNa. We 

 thus see that the formulae (29) and (38) must be identical. 



It has to be remembered that the above formula needs to 

 be modified when both the coils are helical. If the pitch o£ 

 one of the helices be a very small quantity the correcting 

 factor will be small. It is advisable, however, when actually 

 measuring the mutual inductance to investigate whether 

 rotating * one of the helices round the axis through various 

 angles alters the mutual inductance. 



|; 'Even when Legendre's tables are accessible, the evaluation 

 of (38) is laborious. We shall, therefore, give a series formula 

 for M which is convenient in many cases. By the binomial 

 theorem, we always have 



A 



"P sin 2 cos 2 . 



2.4 31 2.4.6 

 where 



* o 



f-' 2 s in 2 * 6 cos 2 e m 



n ~* l-c 2 sin 2 



* o 



= ^-i, ( ff/ sin 2 »- 2 cos 2 e 

 c 2 c 2 J 



(39) 



d0 



Pn'-! 1 1.3. 5 .... 271-3 TT_ 1 



~ c 2 2n , 2.4.6....2/2-2* 2 * c 2 ' ' 



Now it is easy to show that 



p __ it _jr_ ( a + h ) 



1 4(i + VT^?y 16* a 2 ' 



and if we denote P w /Pi by q n we get 



(a + b) 2 1 1.3.5. ...2n-3 a , . 



** = ~Tab~ gn ~ l ~n -2.4.6.... 2n-2 ' b' * ^ 



Thus the successive values of q n can be readily computed. 



Since ~P n is always less than P»-i, and \ is always less 

 than unity, it follows that the series given above is always 

 convergent. The formula for M can. therefore, be written 



* Lord Rayleigh, British Association Report (1899), p. 241. 



