117 
the particles of magnetic bodies, &c. 
whole force with which the magnet urges the ring in the 
direction of its rotation will be represented by 
2 M' 
f^C _ sin, (p __ j_ zr a sm.° (p _I / 
^ I ^c' + 2 r- (i—cos. ?)) I * |c' +2r*(i—cos.<p|*j 
the integral being taken from <p = to = -tt, and ^ M * 
being put for the constant multiplier w ; or if we consider 
the action of two magnets diametrically opposite to each 
other, the force will be represented by double this integral 
taken from (p = %(/ to cp = i tt. The force in the latter case 
will therefore be represented by 
^c' +2r^(i—cos. (c' + zr'^) 
Sin. Ip 
c' + 2r“(i -cos.-ip) 
2 
{77^ 
+ 2 r*' 
{ 2 1 ^ 2 ^ f * 
c> + 2 r*(i-cos.'vP)]' 2 c' (c' + 4 »’*) c' + 2 r*(i - cos.'vj/) 
_ > t I _ c'*+ic'\^ + 6/ ] 
Q 21 ^ '2 42 % L 
c' + 2 r ^ (c' + 2 r^) c' (c' + } 
+ 
c +4cr+i2r 
4 *”0 
, 6 r\d+zr ) 
1" 5 , L 
c' (c' + 4r")^ 
Tan 
-Tan Tani 
If 4 / be a small arc, u extremely small, and d do not 
exceed ^ the first term here will greatly exceed any of the 
others ; and the sum of all the terms multiplied by being 
plus, this will diminish the second term, which is minus : so 
that with these limitations we may consider 
{-;-- V 
* c'*+ 2 r* (i — cos. tp) ^ 
as a very close approximation to the value of the force with 
which the magnets urge the ring. 
In order to obtain a more precise estimate of the value of 
the terms omitted, let us compare this value of the force with 
