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 2 r 2 
/> C sin. a , 2 r o) sin . 2 < 2 > ") j 
• / It* — r*>- d Q> 
+ 2 r 2 (i — cos. <p) j |c' +zr*(i — cos.pj j 
the integral being taken from <p = v|/ to <p — tt, and \ M 2 
being put for the constant multiplier m ^ ; 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 q> = $ to <p = \ «■. The force in the latter case 
will therefore be represented by 
-j c' + 2 r z (l—COS. y]s)j- (c‘ + 2 r z ) 
' Sin. \J/ 
c' -f- 2r 2 (i - cos. •»]/) 
{ — - / + 
2 
{ 2 1 1 2 2 i 4 | - 
c' -f- 2 r a (i -cos.iJ/)j- 2 c' (c' + 4 ?’ 1 ) c' + 2 r 2 (i - cos.-vj/) 
i f i c'*+ 4 c'V 1 + 6 r ) 
q o J 2 1 2 4 2 | f 
c' + 2 r *(<:'+ 2 r*) c' (c' + 4 » a ) 3 
6r‘(c'*+ Z / )_ ( Tan - * tV-M/f ) _ Tan _ ’ f Tan a 4 ) j 
c'(c'‘4 4r 2 f ( C V Sy 
,4 , .8 2 4 
c +4cr4-i2r 
c' 9 (c ,SI + 4 ) 
If 4/ be a small arc, a> extremely small, and d do not 
exceed \ r, 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 
( m 1 1 
* c'*+2r l (i — cos.ij/) * 
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 
