On the Calculation of the Induction- Coefficients of Coils. 63 



diameter in the ratio of v 3 to 2 ; and the fifth term also 

 disappears when the larger coil fulfils the same conditions. 

 Further, if both coils are thus proportioned and placed, the 

 even terms, so to speak, doubly vanish, so that any little 

 inaccuracy in the placing of the coils can only very slightly 

 affect the result. 



We are thus left with the first, seventh, ninth, eleventh, 

 &c. terms of the series. If one coil has half the radius of 

 the other, the error made by taking only the first term in 

 calculating the inductance &c. of the pair of coils is only 

 about 1 in 26,000 ; and if the ratio of the radii is as great as 

 f, only 1 in 4500. 



This result is, it seems to me, of importance both as regards 

 the construction of coils to serve as mutual or self-induction 

 standards, and the choice of the proper arrangement of coils 

 for use in an electrodynamometer for the absolute measure- 

 ment of currents. 



The mutual electrokinetic energy of two circles carrying- 

 unit currents is given by the equation 



T=4*V sin* <£ sin* </>' 2 j^ Z/(c/>) . Z/(f ) . Z,(0) (*)'; (1) 



(p<r) 

 where, as shown in the figure, <£, ty are the angles which the 



radii of the circles subtend at the intersection, c, of the axes, 

 which is taken as the origin of the zonal harmonics ; Z/(<£) the 

 differential coefficient with respect to cos (f> of the zonal har- 

 monic of the ith order for the angle (/> ; Z/ ((/>') the corre- 

 sponding function for <£'; Z*(0) the zonal harmonic of the ith. 

 order in terms of the angle 6 between the axes of the circles ; 

 and /-, p, (p<r) are the distances of the circular arcs from 

 the origin. 



Now instead of two circles take two narrow circular elements 

 of two-single layer coils the axes of which intersect, and the 

 numbers of turns in which per unit of length along the axis 

 are n, n . Let a be the radius of one element, and <r its axial 



