330 Solution of a Problem in Fluaons. 
centre is removed to an infinite distance; .".7 being infinite, the 
c’2cosec.?) : : 
 -, becomes a quantity of the first order relatively to 
c/? 
the other terms, and is hence to be neglected ; and a =Const. 5 
d dy\? 
dr=dy; cot. }= 533 hence F as —d GH (4). Let the equa- 
dy 
: 5? dy\? 62 
tion of the curve be y? = X (2ax—x?) (5) 5 shen 3°) =a x 
é ) (“”) wecinis, «ds 1 
ged ; hence —d de) ary} or F as ye (5) is the 
dy 
equation of an ellipse; make a=, and it is a circle ; change the 
sign of x? and it becomes an hyperbola; neglect x? and it is a pare- 
bola; but the solution which I have given comprehends all these cases. 
(Prin. B. 1. sec. 2. prop. 8. and sch.) By (8) of the Journal (for 
July) when the motion of the particle is wholly in the plane, (7, y) 
rd?x+yd*y . a ie 
ee ee eas ; if the force is directed to the origin of x and y, 
ad? d? , 
daa R= — sd (6). Let y 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 (n= any 
given number; or = sin. 9); let F, r, become F’, 7’, n the changed 
md? i 
curve; .". (6) becomes F’ =—Fgp (0: 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’:ir: 7’ (Prin. B. 1. prop. 10. latter part of the schol- 
“fe : hen lL 
_ ium.) By (M) of the last Journal, I have —34(]) +(d-) je 
=Fadr (8); pu ~=R, suppose dy=const. reduce, and there results 
on re i 
der +R — say. =0 (9); which is a form of F, very useful im 
