330 Solution of a Problem in Fluxions. 



centre is removed to an infinite distance; .'.r being infinite, the 



c'^cosec.^4' 

 term ^ becomes a quantity of the first order relatively to 



the other terms, and is hence to be neglected; and— = const. ; 



dy (^y\ ^ 



dr=dy; cot. 4'=J'; hence F as — ^Ht") (4). Let the equa- 



dy 

 tion of the curve be y^ = — X {2ax -x^){b); then \ j" / — ■ ~7 X 



V" V 5 1^^"^^ -^Kdx) ==0^ 5 or F as -. (5) is the 

 dy 

 equation of an ellipse; make a=b, and it is a circle ; change tlie 

 sign of o:^ and it becomes an hyperbola ; neglect x^ and it is a para- 

 bola; but the solution which I have given comprehends all these cases- 

 (Frin. B. 1. sec. 2. prop. 8. and sch.) By (b) of the Journal (for 

 July) when the motion of the particle is wholly in the plane, (x, y) 



xd^x-\-yd^y 

 F== — -J7^ ; if the force is directed to the origin of x and y, 



xd^y rd'y 



d'^x= .■.F=— --7T2 (6). Let 3/ be changed in any given ra- 

 tio ; or let its inclination to the abscisses (x) be changed from a right 

 angle to any other given angle 9. Then y becomes ny (w= any 

 given number; or = sin. 9) ; let F, r, become F', r', in the changed 



r'd~y 

 curve; .■. (6) becomes F' = — ~3i^ C^)- Hence if two particles of 



matter are supposed to describe these curves so as simultaneously to 

 be at the extremities of corresponding ordinates, I have by (6) and 

 (7), F : F': :r : r' (Prin, B. 1. prop. 10. latter part of the schol- 

 ium.) By (M) of the last Journal, I have — 1^ c? ( (~ ) + \d~j ) = 



dv^ 



=Fdr (8); put -:=R, suppose c??; = const. reduce, and there results 



d'R F 



-^^-{-R. -— j3-2=:0 (9); which is a form of F, very useful in 



physical astronomy. 



