116 
Mr. Christie on the mutual action of 
or, if <p be the angular distance of ds from the point vertically 
over the pole of the magnet, by 
^mir d (p 
The force on d s, or rd<p, in the direction of a tangent to the 
ring, that which urges it in the direction of the rotation of the 
magnet, will therefore be represented by 
4 m i r® sin. Q d<p 
P* 
The intensity z, is, as I have before mentioned, greater than 
that due to the distance p ; let us then suppose that it is the 
intensity corresponding to a position of the magnet at a very 
small distance a behind its position at the instant at which we 
are estimating the force of the magnet on ds. Considering &» 
as a right line, the distance between this point and ds will be 
{ ^ sin, (p — d<>)^+r“(i — cos. cpy j 
+ 2 r* (1 — cos. (p) — 2 reo sin. (p } 
where d represents the vertical distance between the pole of 
the magnet and the middle horizontal section of the ring, or 
p in the first set of experiments. We shall therefore 
have, being a constant multiplier, 
2 _ -- 
c' -f* 2 (i — cos. (p) — 2 r w sin. <p 
or, since a is extremely small, 
. /u., I 2 jj,/ r Cl) siti. (p 
I = -j- -? -3-i a , 
c + 2 r® (i — cos. Ip) -j c' + 2 r® (i — cos. <p) > 
very nearly. 
If then, according to what I have before stated, ri[/ repre¬ 
sents that portion of the ring, behind the magnet, whose 
action is destroyed by the action of the part in advance, the 
