Field of Helical Currents. 44)") 



in the form 



M =" jy ^[ B i{ 1 -2«A 1 -y7jyA 4 -. • •} 



-B 3 {l-i^ 2 2 -^-J^V-...}], (41) 



where R x 2 = {a + b) 2 + fa + 7* 2 ) 2 ; A-! 2 = 4a6/R 1 2 , 

 and R 2 2 = (a + 6) 2 + (Ai - As) 2 ; V = 1«&/R 2 2 . 



This formula gives a complete practical solution o£ the 

 problem. 



Let us take the case of the two cylindrical coils for which 

 a = 5,b= 1, //! = 15, h 2 = 2*5, Ni= 300, and X 2 = 200, mentioned 

 in the preceding section. By (39) we easily obtain 



^ = 0-7, £3 = 0-5525, ^ = 04618, ^=0-3992, 

 2 6 = 0-3527, 27 = 0-3170, &c. 

 Also E 1 2 =(a + 6) 2 +(7i 1 + /z 2 ) 2 = 387-25 ; E 1 = 19'6787 ; 

 ^ 2 =(a + b) 2 +fa-7i 2 ) 2 =237 25; R 2 = 15*4029 ; 

 Ay* =80/387-25 ; V= 80/237-25. 

 Thus, substituting in (11), we have 



M = (47r) 3 7 6 5 10 ' [19*6787 (1-0-07230-0-00295 



-0-00026-0-00003-...) 

 -15-1029(1-0-11802-0-00785 

 -0-00100-0-00020 



-0-00001-0-00001-. . .)] io- 9 

 = 0-0011995 henry, 

 which agrees with the answer found from (29). 

 When 7? 1 = ^ 2 =7j, 

 A^ 2 = ±ab[{ (a + b) 2 + 17t 2 } and 7 2 2 = lab j{a + b) 2 = c 2 . 



In this case the series of terms with the suffix 2 in (11) may 

 converge slowly. It is better, therefore, to express the second 

 half of the formula on the right-hand side of (11) in terms of 

 elliptic integrals. The formula * now becomes : 



M =^» [^{i-JaA'-i^-i^A.-. . . }] 



+ 3^ r ^l(« 2 + ^)(F-E)-2aiF], (42) 



where the modulus of the elliptic functions is 2(ab) 1 - 2 /(a + b). 



* For another formula in terms of incomplete elliptic integrals, see 

 J. G. Coffin, p. 124, I. c. ante. 



