H. A, Rowland—Absolute Unit of Electrical Resistance. 829 
Let N be the total number of windings in the galvanometer. 
Let R and r be the outer and inner radii of the coils. 
Let X and = be the distances of tl:e planes of the edges of the 
coils from the center. 
t a be the angle subtended by the radius of any winding at 
the center. 
Let 5 be the length of the radius vector drawn from the center 
to the point where we measure the force. 
Let 6 be the angle between this line and the axis. 
Let ¢ be the distance from the center to any winding. 
Let w be the potential of the coil at the given point. 
Then (Maxwell's “ Electricity,” Art. 695), for one winding, 
te=—2n{1—c08 a-+-sinta( 20, (a) Q,(0)-+3(5) 2(a)Q()4+-&e.) | 
t 
and for two coils symmetrically placed on each side of the 
origin, 
‘ 1/b\" 5 1/d\*, 
w=47 cos a—sin’a() 3) Q, (2)0+4(3) Q, (2) Q(0)-+&e.) 
where Q, (9), , &c., denote zonal spherical harmonics, and 
Q,’(@), Ss a oO denote the differential coefficients of spheri- 
cal harmonics with respect to cos a. ; 
As the needle never makes a large angle with the plane of 
the coils, it will be sufficient to compute only the axial com- 
nent of the force, which we shall call F. Let us make the 
rst computation without substitution of the limits of integra- 
tion, and then afterward substitute these: 
Fates sa MI ae 
N 
F=3R—opy (X—=) Jt war, 
and we ean write 
oe 22N 2 6 H lik & ° 
F=——yx=a | H,+H,27Q,(4)+H,0Q.+ & 
where H,==2 loge (r+a/z?+r") 
He 13:5. (2é—1) sin? 1 ee ee na ltisen € &e. | 
2 SPIN aA eae ge 2i—1  2—3 
A st 
2i-4 i(¢-1)(i—2 
B=A oe ( )(¢—2) 
‘M—1 = (241) 2 
2i—6 , i(i—1)(i—2) .. (@—-4) 
C=B ast (2i—1)(2¢—3)2.4 
pac 428 i= 8) 
=“ 37—5  (2i—i) (2i—3) (27-5) 2.4.6 
E= &e., &. 
