70 Solution of a Problem in Fluxions. 



is wholly situated J (since x, y, z, are common to s and (Z).) By as- 



suming (l) for the plane oi x, y, I have2;=:0.'.^=0, and t: — 0, and 



(C) does not exist: also (A), (B), become the same as in the last 

 Journal, (on the suppositions there made,) as they evidently ought to 

 be. Put /•sin.-^=p the perpendicular from the centre of force to 



C'_ ds 



the tangent tos at the place of the particle; ^~ih ^^^ veloci- 

 ty ; V = the velocity of a particle describing a circle, about the same 

 centre of force, at the distance r; D= the distance fallen through 

 to acquire V from rest, by a particle, when continually acted on by 



pdr 

 a constant force, equal to F, (at tlie distance r;)--T~— R, half the 



chord of the equicurve circle, with s at the place of the particle, 

 estimated on r. It is evident that (I) can be changed to 



C'^ / 1 \ C'2 / 1 \ Cdp V^dp V2 



--7i-^l rT-zr^r 1 = — 7^-di — — — — 



2 yr^ sin. ^4'/ 2 \p^ / p^dr pdr R 



dr dr 



— — T5^— ,p • — — = F= =arrr(L); and that (H) can be 



C'2 /r^'dv^'i-dr^x /ds-\ VdV 



diangedto--^^^- ^,^^, j = -dlj^j = --^=F jM)', 



dr 2dr 



,-.YdV+Fdr=0 (N). 



If Fdr is integrable, I may change d into S, and (N) becomes 

 V(5V+F^r=0 (o); [S being the characteristic of variations.) I 

 shall suppose that Fdr is integrable ; that is, that F is constant, or a 

 function of r only; and shall put its integral SFdr=(pr. Then the 

 integral of (N) is V^^D- 2(pr (1); (D= const.), (1) corrected is 

 y3_Y//2_|_2((p/.;._(p^) (2); V, cp'r, being the values of V, cpr, at 

 some given point of s. 



From (2) it appears that V depends on V", (p'r, cpr; suppose then 

 two spherical surfaces to be described from the centre of force (as 

 centre) through the distances corresponding to cp'r, <pr; then it is evi- 

 dent that if the particle passes through the first of tliese surfaces (in 

 any point whatever) with the velocity V'', it will always arrive at the 

 second with the same velocity, V, and that in whatever point it may 

 meet it, and whether it moves in a straight line, a curve, or curved 

 surface ; provided the curvature be continued, so that the direction 



