342 Central Forces. 
Art. XIV.—On Central Forces; by Prof. Turoporr 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 7; also by a force f’ which aets at 
right angles to 7, in a plane passing through 7 at right angles to the 
fixed plane, this force tending to diminish 6. ‘Then by resolving f, f’ 
in the directions of 7’ and z, I have fcos.d, —/f’sin.é for the forces in 
the direction of r’, and fsin.é, f’cos.4 for the forces in the direction 
of z. Hence the whole force in the direction of 77 =P=fcos.d— 
j’sin.6; also the whole force in the direction of z=S=fsin.d+/’cos.6 5 
but since s=tan.é .”. cos.d= ? Af al hence P= 
/1+s? /1+s? 
Nae en See =(sf+f). If Q is such a function that 
dQ dQ dQ al dQ dQ 
ah ~ pio dys ther Pas ha a): se 
1 dQ. dQ ; eye 
vite alata Ys but since rcos.d=r aie Tos 
1 dQ d 1 
u V1i+s?\Vi+s? dr V1+s? 
d d d 
eee oe 7)s also te Now since Q is a func- 
tion of 7, v, 6, and as these are functions of w, v, s, .*. Q is a function 
: d d d d' 
of u,v, s; hence pert geeet ie ee ae or 
dQ. Pera de mee, 
q-ar+ a De dqutE ds; but u=¥—7* , s=tan.d give du= 
Hd toner tes u? dr 
dj 
ae eo 
rr/ 1s? r2 J1+s cos.76 
= d}(1+s?); by 
dQ. dQ 
substituting these values of du, ds, there results ee 3 O=F, 
u?dr 
Vits? 
tion; .°. by comparing the coefficients of dr, dd, I have wr 
uz? dQ dQ dQ iQ z 
~Vits? au’ de ic tals lad e = ; hence P=usz twa 
dQ 
(usd — +20 -+s*)d3, which must be an identical equa- 
