6 Mr. Christie on the effects oj temperature on 
The equation L = o is in this case, 
r 
0 
and consequently ^ z= 2 jr, = 2y. 
The equation ( 2) therefore becomes 
Xy — Y X = 0. 
Substituting in this equation the values previously found for 
X and Y, and dividing by y, we obtain 
M-~F. 
[(R- 
p) • (: 
+ 
I 
0 
^ ^ \{SV (S<r,y 3 ^ 
Let (f> be the angle which the axis of the needle makes with 
the meridian, or the azimuth of the point of equilibrium, and 
we shall have, 
[sOj,]"— (R— ?)”+'• — 2 '•(R — p) cos. <p ; (R— py+ r^+ 2 r(R— p) 
(j/^y=(R+p/+''— 2r(R + p)cos. ip; (R + p)’+ r’‘+2r(R + p) 
Substituting these values in the equation (A), it becomes, 
R-f .. ! 
(A) 
cos. (p; 
cos. (p. 
M — F.< 
t 
2r(R— p) 
(R— 
cos. <p 
-i-l 
R + p 
I (R + !>)•+ I 
+ 
2 r (R + p) 
\ i i,i2r(R4-p) 
( (R+pr+^ 
^ = 0 
(B) 
(R+p)*+r 
From this equation the value of F in terms of M may 
be found for any values of <p, the distances R, r and p 
being known ; and if we suppose M constant during the 
observations, the variations in the intensity of the force F 
may be obtained from the observed variations in the value 
of (p. 
If the angle <p does not differ from a right angle by more 
than 10° or even 20°, by expanding the several fractions, no 
sensible error will arise by limiting the series to a few of the 
