M= 



60 Mr. J. H. Poynting on the 



circular circuits are on the same axis at a distance apart greater 

 than the radius of the coils, the following formula is obtained. 



Let a = distance between centres, 

 b = radius of either circle, 



c = distance of either circumference from centre of 

 other, 

 M = coefficient of induction. 

 Then 



4tt 2 5 4 f 1 3Z) 2 15^_35^ 2835^ „ m 

 c 3 \2 4c 2+ 8 c 4 8 c 6 + 256 c 8 &C * W 

 or 



_47r 2 6 4 ri_3^ 75^_490^ 6 24570 Z> 8 



a 3 \ 2 2 a 2 + 16 a 4 32 a 6 + 256 a 8 W 



Of these the latter uses directly the distance between the 

 centres, the observed quantity — but is not nearly so conver- 

 gent as the former, in which c may be at once deduced from 

 c = \/rf + W. 



To obtain formulae which might be strictly applied to the 

 sonometer, we should have to consider the more general case 

 of two coils of unequal radii b and /9, for which I have found 

 the formula corresponding to (2), viz. 



4wgg /l 3 5 2 + /3 2 15 tf + Ztfp + ff* 

 a 3 V2 4 a 2 + 16 a 4 



35 6 6 + 6£ 4 /3 2 + 66 2 /3 4 + /3 6 , , /QN 

 "32 ? +&C * (3) 



We should then have to take the finite integrals of each term 

 between the limiting values of b and j3. But this would be 

 exceedingly complicated and would require a knowledge of all 

 the details of construction ; and we may at least get a first 

 approximation to the true result by replacing the coils by a 

 single one of a radius intermediate between the greatest and 

 least radii. 



In Prof. Hughes's paper (Phil. Mag. July 1879) he gives 

 the internal and external radii of his coils as 15 millims. and 

 27*5 millims. respectively. I have considered, then, that 

 25 millims. will give results not very far from the truth ; and 

 as it makes the calculations considerably easier, I have taken 

 that as the value of b and applied the formulae to the numbers 

 given in the paper. The resultant current in the middle coil 

 was zero when it was distant 47 millims. from one end and 

 200 from the other. This enables us to find the ratio between 



