Determination of Resistances in Absolute Measure. 331 



with certainty by thermometers, would influence the result by 

 as much as one part in a thousand. 



If it be granted that the comparison of currents and the 

 reference to the standard of resistance can be effected satis- 

 factorily, we have only to consider the amount of error involved 

 in the determination of M, the coefficient of mutual induction 

 between the two circuits, which is the fundamental linear 

 measurement. If the two coils are of very nearly the same 

 size, it appears from symmetry that the result is practically a 

 function of the mean of the mean radii only, and not of the 

 two mean radii separately. It is also of course a function of 

 the distance between the mean planes b. Leaving out of con- 

 sideration the small corrections necessary for the finite size of 

 the sections, we consider M as equal to 4:ir\/Aa multiplied by 

 the function of 7, given in tables appended to the second 

 edition of Maxwell's ' Electricity,' where 



%s/ka 



or, if we identify A and a with their mean (A ), 



, 2A 



tan 7= — — . 

 b 



The error in M will depend upon the errors committed in the 

 estimates of A and b. If we write 



dM. . dA . db 

 M =X A7 + ^P 

 then, since M is linear, 



Thus, if b were great relatively to Aq, 

 X=4, f&=— 3, 

 a very unfavourable arrangement, even if it did not involve a 

 great loss of sensitiveness. The object must be so to arrange 

 matters that the errors in A and b do not multiply themselves 

 unnecessarily in M. But since p is always negative, X must 

 inevitably be greater than unity. 



The other extreme case, in which b is very small relatively 

 to A , may also be considered independently of the general 

 tables; for we may then take approximately (Maxwell's ' Elec- 

 tricity,' § 705) 



M = 47rA log|^-2}, 



whence _ _ 1 



*- log(8A /B)-2' 



Z2 



