the intensity of magnetic forces y &c, 5 
trary name at the unity of distance, be F : then the forces’ 
acting on the south pole of the needle will be, 
M in the direction x\ 
F F 
— r in the direction • r-» in the direction ; 
', in the direction s v . 
r* F 
— in the direction s o-q; r, 
The north pole of the needle will be urged by forces equal 
and parallel to these, but in contrary directions ; so that in 
investigating the conditions of equilibrium of the needle, we 
may consider only the equilibrium of the south pole urged 
by forces double of the preceding, and constrained to move in 
a circle ; and it is evident that the equation of equilibrium 
will be the same, whether we take these forces, or tlie doubles 
of them. 
9 
Resolving these forces into others in the directions x and jy, 
calling X the sum of all the forces in the direction a?, and Y 
the sum of all the forces in the direction y, we shall liave. 
X = M 
Y=:F. s 
SR-p 
X 
R + f — X __ 
R 4- p 4- .T I R 
P 4- 
3 S 
y 
\ 
The general equation of equilibrium for a point acted upon 
by forces in the same plane, and constrained to move in a 
curve whose equation is L~o, is 
X r/x “h Y X d L =0. (1) 
From this we obtain 
^ ^ = 0, and Y + X ^ = 0 : 
dx 
X . ^ 
dy 
Y d L 
•rf7 = « 
(«) 
whence 
