* 
2. Central Forces. 
Arr. XIV.—On Central Forces ; by Prof. Turopore Srrone. 
(Continued from p. 135.) . 
Ler the particle be acted upon by T as before, and by a force f, 
which is directed to the origin of r; also by a force f’ which acts at 
right angles to r, in a plane passing through 7 at right angles to the 
fixed plane, this force tending to diminish 4. Then by resolving f,f’ 
in the directions of r’ and z, I have fcos.4, —f’sin.4 for the forces in 
the direction of 1’, and fsin.d, f’cos.d for the forces in the direction 
of z. Hence the whole force in the direction of 7 =P=fcos.d— . 
J'sin.d; also the whole force in the direction of z=S=fsin.d+f'cos.d5 
1 
but since s=tan.é .". cos.d= ; ns es hones Be 
f/1i+s? /1-+s? 
1 1 
: Jiaae 9) seooy ry Beds If Q is such a function that 
dQ dQ_, dQ 1 Qe 
Tye Ph ST aac ik Pao > then Ee Fi (—i- EP s= 
oe eee 
Vitis? PS Te bat 3 but since rcos.d=r =: ere 
eo a lf 
aa hence P= (a di ae Ss= vide 
u dQ\ - ee d 
a Bae): also (eae rs Now since Q is a func- 
tion of r, v, ¢, and as these are functions of u, v, s, -°. Q isa function 
= d d d d dQ: dQ 
of u, v, 83 hence 90 11 int Raa Pint pao Ge or 
dQ, dQ. dQ, dQ sortie 
get Gea geet G esi but u=V1T stand give du= 
sds_ drs/1-+-s? “ utdr a 
VIS" — usd) =» ds = —_,, = (1487) 5 b 
rf 1+s? ee /i+s? as oos.88 — es 
d 
sd d 
substituting these values of du, ds, there results = dr+ R= Ta 
a u?dr dQ : ech <a 
usd) —-————) + =— 2)¢ oT ae 
( JEL +q, (14+s?)d!, which must be an identica eq 
tion; .". by comparing the coefficients of dr, d}, I have G-= 
uz 
pea S| dQ dQ dQ dQ aQ 
Fie aa da =e +427) g, 3 hence Peasge tT 9 du’ 
