286 Solution of a Problem in Fluxions. 
dv? —d*r 
ha F(A’) and (B) be- 
- 
not exist ; also (A) becomes 
comes d (Gr =I", (B’). These results can readily be 
dt 
found from the equations r?=#*?-++y? . . ‘= tan. v, by the 
same method as before. Again, if a =0, I have r?dv=c'dt 
c ; 
(G), (c’= const.), hence dv?= —{—5 substitute this in 
esr 
(A’), and there results —=-— aaa (D), multiply by dr, and - 
— sail to r, and reduce, and there results 
c'dr 
== 5 aa feoning ns Oe eee ‘2 —9r?SFdr 
(E), which agrees with Laplace’s result, (Mec. Cel. Vol. 1. 
p- 113,) and is same as that of Newton, (Principia, Vol. I. 
ioe Vill. prop. 41.) 
The equation (D) may be put under the form 
=" r* yp? 
aie oe a) =F’, substitute for dt? its value ar andit 
~ Bade 
becomes 73-9 4 (Fie a) =F (H), which agrees with 
(4) of Laplace, at the slabs before cited; (H) can also be 
42 t.? 
put under the form; ~Z-d = rn) =F (1), (¥ being the 
angle at which the radius vector r petits the curve and — 7 
its cotangent. By substituting in (A’) for dt? its value as 
rdv? —d?r 
given by (G), I have (2)" =F (K), or F varies as 
rdv? —d?r 
= ae (since for a given centre of force in a givel 
curve, c’, is constant) which agrees with Newton’s result, 
eee tat, sec. second, prop. 6. cor. Ist. his QR being the 
