414 
LORD RAYLEIGH AND MRS. H. SIDGWICK ON THE 
are subject to the relation 
df dk. da 
7 = X A +l*a + 
dx 
v — 
X 
X+p, ~\~v — 0 
(2) 
( 3 ) 
If the coils are placed at such a distance apart that the attraction is a maximum, 
v=0, and the calculation is independent of small errors in the value of x. Under 
these circumstances X+/x = 0, so that proportional errors in A and a affect the result 
in the same degree and in opposite directions. In other words, the attraction becomes 
practically a function of the ratio a /A only. 
To this feature we attach great importance. The ratio of galvanometer constants 
can be accurately determined by the purely electrical process of Bosscha -without 
linear measurement of either, and from this ratio we can pass to that of the mean 
radii by the introduction of certain small corrections of the second order. 
In this way all that is necessary for the absolute determination of currents can be 
obtained without measurements of length, or of moments of inertia, or even of 
absolute angles of deflection. The forces are, however, evaluated in gravitation 
measure, so that the final result requires a knowledge of gravity at the place of 
observation ; but except through this quantity there is no reference to the units of space 
or time. 
§ 6. The final calculation of the attraction is best made with the use of elliptic 
functions ; but useful information, sufficient for a general idea of,the conditions and 
for the design of the apparatus, may be derived from the series developed in Max¬ 
well’s ‘ Electricity,’ § 304. If B, 6 be the distances of two coaxal coils of radii A 
and a from a point on the axis taken as origin, and C~ = A' + B~, we have 
tfM 0 AV 
= 77 
db 
C 4 
-p T>2 
1.2.3- + 2.3.4- C3 
iA- 
6 + 3.4.5 
B(B 2 —jA 3 ) 
C 5 
(b»-±a>) + 
. . (4) 
in which a, 6 are supposed to be small relatively to A, B. If we limit ourselves to 
the first term, which we may do when aj A is small, we see that so far as it depends 
upon the small coil the effect is proportional to the area. The position of maximum 
effect for given coils is found by making B/C 5 a maximum, which leads to B = ^-A; so 
that to obtain the greatest attraction the distance of the coils must be equal to half 
the radius of the larger. 
In the present measurements there were two equal fixed coils, one on either side of 
the small coil. If we take the origin midway between, the terms of odd order in 6 
ultimately disappear in virtue of the symmetry, and we may write 
d M 0 A 2 « 2 f 
db — 7r 'c r j 
1.2.3 ? +3.4.5 
B(B 3 
( 5 ) 
