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.é, put z= the perpendicular from the parti- 
cle to the fixed plane; then by the theory of accelerating forces (or 
22 dz? 
by (a) Vol. XVI, p. 284,) qe tS=% or a( Za) +S=0, or substi- 
Qdz 
4¢/2 dz? 
for di?, it becomes dere) +S=0; but 
r4dv2 dv? 
2 use’? 
tuting 
z=48 = Gz cia ce and u2dz=uds —sdu; hence 
2(uds—sdu)?\ 2 
d (Aaa) Fines =A which, by making dv constant and sub- 
uds —sdu : 
du 
sat ‘ Tie eg d2szr8 (du Ta 
stituting for cde’ its equal [5% gives aa aX (agit) + 
ds 
Tit 
Cus 
$ d? 
=0, which by substituting the value of a and that of ¢’* 
d. > 
dts T7+S8 — Ps 
becomes 7348-7 Tdn\’ (9). Hence I have d¢= 
oles) 
u 
T>-—Pu 
dv d?u dv d?s 
Lee 
u? ht +2f 5 u? i842) 
ds 
Tatars : eet 
2 (C); which are sufficient to determine the 
u? (is 4: 2f. org ) 
place of the particle 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. 
