Central Forces. 135 
fixed plane. In this case, a resolved part of the force towards the 
first centre and of the disturbing force act at right angles to the fixed 
plane; let S denote their resultant, which I shall suppose acts towards 
the plane. Let s=tan.4, put z=the perpendicular from the parti- 
cle to the fixed plane ; ee ne the theory of ob welaelaie forces (or 
by (a) Vol. XVI, p. 284,) Gos ge :0, or ae )+8= O, or substi- 
dz 
usc’? dz2 
for dt?, it becomes dae +S=0; but 
Qdz 
r'4dv2— dv? 
tuting ya Tyree C2 
s uds—sdu 
z=rs=> pn aa and u°dz—=uds — sdu; hence 
(—“S c’? (uds — sdu)* 455 
do™ —, =0, which, by maton dv constant and sub- 
uds —sdu 
ait 
dv 
dy? c/2 43 
mee 
stituting for c’de’ its equal [3° EINES. saiiy is 
d?s $ (a3 )+ 
d? 
—,- =0, which by substituting the value of = and that of c/? 
d 
T+S8—Ps 
d2 
becomes ve. a ——,2 (9). Hence I have di= 
dv? Tdv 
w(ie+2f5) 
—7 PG» (C)3 which are sufficient to determine the 
wlitt of) 
place of the 1 ah at any given time. The equations (A), (B), 
(C) are the same that Woodhouse has found in a very different man- 
ner at p. 95 of his Physical Astronomy; they can easily be obtained 
from the equations (A), (B), (C) given in the Journal, Vol. XVI, 
pp- 284, 285; but I have preferred the above method, as being in 
some respects more simple. 
