7G2 Dr. Fleming on the Determination of the 



M of such circuits, viz.. 



-P? 



1= -^cos^ 



where r is the distance of any pair of elements ds and ds' of 

 the two circuits and 6 their inclination, we" extend the inte- 

 gration over every possible pair of elements in two rectan- 

 gular circuits. It is not difficult to arrive at the following 

 result for the mutual potential energy of the two rectangular 

 circuits conveying unit Currents, viz.* 



M 



L ° (S4V!3 2 + S'* + ^ - ° S' + ^S /2 + 8 2 + 6 2 > 

 + 2 ^S 2 ^^ 72 ^ 2 - 2 yS 72 +P - 2 A /S 2 T6 2 + 2*1 • 



To obtain the inductance of a rectangular-shaped circuit, 

 we have then to obtain the mutual potential energy of two 

 rectangular circuits conveying unit currents placed at a 

 distance apart equal to the geometric mean distance of all the 

 filamentary currents into which we can suppose the real 

 current in the real circuit divided. If we consider that 

 circuit to be a circular-sectioned wire and the currents to be 

 high frequency, then, as Maxwell shows, the geometric mean 

 distance of all points lying on the circumference of a circle 

 is equal to its radius f. 



Hence, in the above formula, if we make b small compared 

 with S or S', and substitute for b the value d/2, where d is 

 the diameter of the round wire, we shall have an expression 

 for the inductance L T of a rectangular-shaped circuit of such 

 wire for currents of infinite frequency. The expression is 

 as follows : — 



L 1 -4[(S + S / )log^'-Slog(S + N /S 2 T^ 72 ) 



- S' log (S' + */&TW 2 + 2y/ & + W 2 - 2 (S + S')] (1) 



The above formula is a strictly accurate one for infinite 

 frequency, and can easily be applied to any case of a real 

 rectangular circuit. The logarithms are of course Napierian. 



* There is no need to give the various steps of the integration as 

 substantially the above formula is given in Mascart and Joubert's 

 Treatise on Electricity and Magnetism, Atkinson's English Transla- 

 tion, vol. ii. p. 477. 



t See Clerk Maxwell's Treatise on Electricity and Magnetism, 2nd 

 ed. vol. ii. p. 298. 



