of the Induction- Coefficients of Coils. 69 



Since the currents in the coils are each unity, the expres- 

 sion in (13) is the coefficient of mutual induction M of the 

 two coils. By constructing the coils in the manner here 

 specified, and placing them so as to be concentric and coaxial, 

 Z^d) = 1, and the coefficient of mutual induction is, if «^> 2a/3, 

 practically given by the equation 



M=8wWa»?^, (14) 



where x 2 , I2 are the half-lengths of the coils. 



Accurate standards of mutual inductance could thus, I 

 venture to think, be very conveniently constructed. 



Equation (13) gives of course also the coefficient of self- 

 induction of a coil. It is only necessary to make the coils 

 equal in size and coincident and take the value of T so given 

 as the required coefficient. The first term will not in this case 

 suffice for so high a degree of approximation, although the 

 series is still fairly convergent. 



The application of these results to the construction of abso- 

 lute electrodynamometers is also obvious. By making one 

 coil small enough to be suspended concentrically within the 

 other, but not so small as to render the exact measurement of 

 its dimensions difficult, we can construct an instrument the 

 constant of which is easily calculated with great accuracy. 

 The couple © turning the suspended coil would then for unit 

 current in each be given by 



X &Z 



n 



sin (9 (15) 



Should, instead of single-layer coils, coils of several layers 

 be employed, the channels in which the wire is wound might 

 be so shaped as to cause each layer to fulfil as nearly as pos- 

 sible the ratio of length to diameter stated above. This might 

 be done by making the ends of each channel segments of a 

 cone the vertex of which is at the centre of the coil, and the 

 semi-vertical angle of which is tan _1 \/3/2. Then, by calcu- 

 lating for all the different pairs of layers which can be obtained 

 by taking one layer in each coil, the energy of the arrange- 

 ment and the action of one coil on the other might be found. 

 The accuracy of such an arrangement would of course be 

 limited by the fact that if one layer (as would always be 

 arranged) fulfilled the required relation of length to diameter 

 with an exact number of turns, the rest might only more or 

 less closely approximate to such fulfilment. There would also 

 be uncertainty as to the distribution of the wire, which would 

 be more or less irregular. 



