436 Mr. A. Russell on the Magnetic 



Therefore 



M= 2^ f» C* ar(a-r cos 0) f ^ + A2}1/2 



-V>2 Jo Jo ^ 



-{(Ai-/i 2 ) 9 + A*}i/3]^r. . . (29) 



It can be proved mathematically that this formula is equiva- 

 lent to the following-: 



-R^l-^sin 2 ^) 1 / 2 ]^, . . (30) 



where c 2 = 4aft/ (a + ft) 2 ; R x 2 = {a + ft) 2 + (7 t L + A 2 ) 2 ; 

 B 8 , = (a + ft)*+(J 1 -*,)«; ^^^aft/Rx 2 and £ 2 2 = 4aft/R 2 2 . 

 As we give a proof of the identity of (29) and (30) in the 

 next section, it is unnecessary to give a mathematical proof 

 here. 



It is not difficult to find a series for M by expanding the 

 radicals in (29) by the binomial theorem and then integrating 

 the terms. The author has found several series formulae for 

 M. The following two approximate formulae deduced from 

 them are simple, and when hf + h^ + a* is greater than 

 ft 2 -J-2aft they give accurate results. The first formula given 

 (31) is much the more accurate, especially when l\ x and h 2 

 are not very different. 



M= 2_7rN 1 N 2 (7rft 2 ) f 2d 



{ 



d \ (d 2 + 2AA) 1 / 2 + (d 2 - 21 h h 2 ) V* 



3a 2 ft 8 5a 2 ft 



and 



27rN 1 K 2 (7rft 2 ) r .W+4?^' | , 



1V1- g jl + g^ j, .... (32) 



where d 2 = 7i L 2 + 7i 2 2 + a 2 . 



Let us suppose, for instance, that Nj = 300, N 2 = 200, a = 5, 

 7> = 4, 7^ = 15, and A 2 = 2*5, the lengths being measured in 

 centimetres. Formula (31) makes M = 0*0011995 henry, and 

 (32) makes M = 0*0011992. By the complete formula given 

 in the next section M = 0*0011995. It will be seen that the 

 accuracy of the approximate formulae in this case is quite 

 satisfactory. 



From considerations in connexion with the magnetic 

 potential of a cylindrical current sheet, Gr. F. C. Searle* has 



* G. F. C. Searle and J. K. Airey, ' The Electrician,' Dec. 8th, 1905. 



