Field of Helical Currents. 445 



axial length, will be approximately equal to the self- 

 inductance L o£ the coil. Hence, by (44), we find that 



■where X is the number of turns of the helix and a/(a 2 + ]i 2 ) l < 2 

 is the modulus of the elliptic functions. 



From a paper by Rosa *, the author has recently discovered 

 that this formula has been previously given by Lorenz f . 

 Before finding this out, however, he had given the formula 

 to his students, and several of them had constructed standards 

 of self-inductance by simply turning cylinders of well-seasoned 

 teak in a lathe and winding cotton-covered wire round them. 

 The agreement between measurement and calculation was 

 found quite satisfactory. 



When h is less than a, we get by the Binomial Theorem 

 and (1) and (2), 



L= ^ X2 hi-2 + £( lo «i + l)]- ■ (47) + 



This is in exact agreement with a formula previously given 

 by Lord Bayleigh (' Scientific Papers,' vol. ii. p. 15). 



When h equals (3/4)<2, the number found by this formula 

 is too small by about the quarter of one per cent. For 

 smaller values of h the numbers found by (46) and (47) are 

 in practical agreement. When li is small, a small percentage 

 error in determining it introduces a large percentage 

 error into the calculated value of L. Hence, in making- 

 standards, it is advisable to have h not less than 2a. In this 

 case we can use the following remarkably simple formula : 



When h = 2a, the inaccuracy of the formula (48) as com- 

 pared with the more accurate elliptic integral formula is less 

 than 1 in 8000. If li be greater than 2a, therefore, it is 

 sufficiently accurate for all practical purposes. 



It is convenient to write 



L = P{a 2 jli)S% (49) 



and Tabulate the values of (3 for various values of lija : — 



* Bull. Bureau of Standards, p. 162, Aug. 1906. 

 t Wied. Ann. vii. p. 161 (1879). 



j The much more elaborate formula given by Coffin (/. c. ante, p. 113) 

 may be deduced in the same way from (46). 



