796 PROCEEDINGS OF THE AMERICAN ACADEMY. 



was taken and a calculation similar to the one above for the general case 

 made. This completely verified this term, and proved that the expres- 

 sion used above equation (21) is actually valid for small arguments as well 

 as large. This last expression 2/3 d/i\ multiplied by Zoo = 4 7r 2 N' 2 r^ is 



Zo-Zoo =l^N 2 ri d (25) 



o 



and is therefore the additional self-inductance due to the field in the 

 wires themselves for steady values of the current, or, in other words, 

 this is the maximum possible change in self-inductance. Calculations by 

 means of (25) show this expression to give, for coils of finite length and 

 wires with round instead of square cross-sections, the values : 



TABLE I. 



Comparison of the Theory with the True Change in Self-Inductance. 

 Coil Length. True Change. Value of 2/3 d/^. Ratio. 



46.0 cms. .000380 Henry .000280 Henry 1.35 



30.5 " .000200 " .000160 " 1.25 



15.0 " .000067 " .000057 " 1.17 



13.0 " .000051 " .000044 " 1.16 



The column headed " True Change" was computed by taking the differ- 

 ence between the self-inductance of a coil of mean radius a, computed by 

 means of an exact formula, and that of the same coil, using as a mean 

 radius a — p, where p is the radius of the wire. These results, consider- 

 ing the assumptions made in the theory, are in good agreement with it. 

 To make the theory fit to actual facts the second term in the parenthesis 

 in equation (19) should be multiplied by a constant, the average value 

 of which is approximately 1.25, deduced from the above table. "We 

 should expect some multiplier to be necessary because a current sheet is 

 the equivalent of square wires without insulation, while, in the coil for 

 which the calculations were made the wires are round and have insulation. 

 We may then write 



sh=9- < 26 > 



2 d ( 3 sinh x - 



3 r x \x cosh 



In the proposed use of this theory, however, it is not necessary to em- 

 ploy this constant whose value at best is not satisfactorily determinate. 



As cosh x and sinh x increase without limit as x increases, equation 

 (26) may be written, for large values of w, 



