Solution of a Problem in Fluxions. 329 



Art. VII. — Solution of a Problem in Fluxions ; by Prof. Theo- 

 dore Strong. 



(Continued from p. 73 of this Volume.) 



TO PROFESSOR SILLIMAN. 



New Brunswick, Nov. 9, 1829. 



Dear Sir — I send you the following continuation of my paper. 

 Yours respectfully, T. Strong. 



C W 1 \ 

 The same notation being retained; the form F= — -^ c?l — I 



dr 

 applies easily to the case in which the particle describes an ellipse, 

 the centre of force being at the centre. Let a, p', be the same as 

 at the 72d page of the last Journal : then by what was there found, 

 (since ^=half the sum of the perpendiculars from the foci to the tan- 

 gent at the place of the particle ; the distance of its foot from the 

 point of contact being half the difference of their distances from the 



a^p' d^r 



samepoint);Ihave a^+a/-— i-=r3(l); .•.F=-^,(2); or F 



as r. By changing (1) into a^ —op' + — r = ^^j the ellipse be- 



P 



comes an hyperbola, the centre of force at the centre ; and F as 



— r ; hence it is repulsive. Substitute in (2) for c'", its equal 



r'^dv'^ 



-"T7r~ '•> change the ellipse into a parabola by removing its centre to 



an infinite distance ; which makes r parallel to a ; — = 1 ; put 



rdv 



-j-=y the velocity parallel to the ordinates to the axis =const. and 



(2) becomes in the parabola F = — ^;-=const. (Prin. B. 1. sec. 2. 



prop 10. and sch.) The form (I) of the Journal (for July) is easily 



adapted to the case of the particle describing a curve, when acted on 



by a force parallel to the ordinates (y) which are perpendicular to 



c'"cosec.^4> 

 the abscisses {x). For (I) can be changed to — ^ ~ 



c'2 (cot. "■^) 



—d — -1- == F (3); since the force acts m parallel lines, iti 



Vol. XVII.— No. 2. 15 



