286 Solution of a Problem in Fluxions. 



rdv^ —d'^r 

 not exist ; also (A) becomes ^ =F, (A'), and (B) be- 



(r^dv\ 

 —T— \ =F', (B'). These results can readily be 



dt 



y 



found from the equations r-=x'^-\-y'^ . . -= tan. v, by the 



same method as before. Again, if F'=0, I have r^dv=c'dt 



c'^dt"" 

 (G), (c'= const.), hence au 2= — —i substitute this in 



c'2 d^r 

 {A'), and there results — — -^=F (D), multiply by dr, and 



integrate relatively to r, and reduce, and there results 



rdr c'dr 

 dt= , — .hut c'dt-r^dv,.'.dv- . — 



(E), which agrees with Laplace's result, (Mec. Cel. Vol. I. 

 p. 1 1 3,) and is the same as that of Newton, (Principia, Vol. I. 

 sec. viii. prop. 41.) 



The equation (D) may be put under the form 

 c'2 (dr'^\ r'^dv^ 



—J —d\-Tj^ I =F, substitute for dt^ its value — j^—> audit 



becomes -^fz ~- 'Y ^\V^'^) ^^ (^)' ^^•^^^ agrees with 



dr 

 (4) of Laplace, at the place before cited ; (H) can also be 



put under the ^oxvi\—^~'^-d\—^-J =.Y (I), (4 being the 



dr dr 



angle at which the radius vector r cuts the curve and ^^ ~ 



its cotangent. By substituting in (A') for dt^ its value as 

 rdv" — d^r 



given by (G), I have (^) ' ==F (K), or F varies as 



vdv ^ d"T 



^-T-^ — ' (since for a given centre of force in a given 



curve, c', is constant) which agrees with Newton's result, 

 (Prin. 1st, sec. second, prop. 6. cor. 1st. his QR being the 



