416 ME. DUGALD MTQCHAN ON THE DETERMINATION OF THE NUMBER 
Let f, I' = depth of the section of the coils in the plane of the coils, 
j/=breadth of section of coils perpendicular to the plane of the coils, 
then the effective component of the force due to one of the coils is 
~ 2 %nr ^ (2 — 15 sin 2 a cos 2 a)+^^i (15 sin 2 a cos 2 a— 3 sin 3 a)j, 
(r 2 + Z» 2 ) 
and that due to the other 
2 tt nr n 
c . 7 ^ ^ ---i|l + 2 1 ! pi (2 — 15 sin 2 a' cos 2 a')+T 4 p* (15 sin 2 a' cos 2 a' - 3 sin 2 «')[> 
(d 2 + 5 2 )* 
For the left coil, 
outer radius=16 902 centims., inner radius= 15*312 centims., 
r=16Tl centims., | = 1*59 centim., t?= 2*00 centims., 
. v b 
sina= — cosa = and the correcting factor= ’9998234. 
(r 2 + 6 2 )* (r 2 + 6 2 )* D 
For the right coil, 
r'=16’24 centims., |'=L856 centim., j/=l*95 centim., and the correcting factor 
= •9997692. 
Calling these factors F and F', we have for the magnetic force at the centre of the 
suspended coil due to the two fixed coils when a current of strength C is passing through 
them, 
fJW 2 ] |_ 2 toV' 2 | 
.l(r 2 +//CJ l(P 2 + £ 2 )fJ 
c. 
( 1 ) 
The magnetic moment of the suspended coil is equal to the product of the strength 
of current and the sum of all the areas enclosed by the wire. 
Let c = a mean radius of the suspended coil, i — number of turns of wire in the sus 
pended coil, 
then C . «Vc 2 = its magnetic moment, and the couple due to a current C tending to turn 
the suspended coil round a vertical axis through its centre is 
C 2 ivc 1 . 
( 2 ) 
The outer radius of the suspended coil was 5’240 centims., and the inner radius 3-259 
centims. The difference of radii, L981 centim., being considerable in comparison with 
the radii themselves, the quantity ?Vc 2 was determined experimentally, after the obser- 
vations had been completed, by a method to be afterwards described. [The result then 
obtained is used in all the calculated values of v, except those of 1867 and 1868.] The 
mechanical value of the deflecting couple (2) is given by the equation ^r 2 +g/0=O, in 
E 
which g,=^A_, Wk 2 being the moment of inertia of the coil round a vertical axis, and 
E the elastic couple per unit angle. 
