332 Solution of a Problem in Fluxions. 
parabola, which gives V : V/::4/2 : 1 (11); (P- prop. 16. cor. 7.) 
Since (10) and (11) are independent of the parameter of the conic 
section, by supposing the parameter to vanish the sections become 
right lines, and the particle describes a right line, as stated by New- 
: me, 
ton, B. 1. sec. 7. propositions 33 -- 34, and V ; V’: -*/AC? ay 
as in prop. 33, for in his fig. (1) 2a -r=AC, in fig. (2)2a+r=AC, 
AB 
in both a=—Z— ; but in prop. 34.V2V' 224/23 1 
I will conclude this paper with some remarks on (6) which pre- 
sented itself in the differentiation of (1) of the Journal (for July). 
F F 
--It may be changed to (Sia Jor (St) ot 
d?z 2¥ 
+(; dé == Je=0 (a). If the particle is free, that is, if it is not 
d to move on any given line, or surface ; then , y, 2, are in- 
dependent of each other; and their poediclonis. or the quantities in 
(4) within the parentlieses must each =0, in order that (a) may be 
diy aE d*y yF d?z af. 
mdefinitely tne. Hence 7>=——- os 
which are the equations (g) assumed at page 69, of the last Journal, 
from the ordinary methods of decomposing forces. Again (5) can 
wd? z+-yd?y+2d*z 
di? 
be changed to 
rete ce 
+Fr=0 (ce) ; similarly I shall have 
+ F’r’==0 (c’) should the particle be acted on 
by any Bi: force, F’, analagous to F; a’, ¥, 2’, being r respectively 
parallel to x, y, z; their origin being at the centre of F’; also 7” is 
the distance of the centre of F’ to the particle. Also I shall have 
for another force F”, an equation of the same form by accenting 
@, yy &c. twice; &e. for F” to any number of forces whatever 
In order to teats the combined effect of F, F’, F”, &c. on the pat- 
ticle, I take the sum of the equations (c) &. then I suppose the ce 
sition “gee vey Sora infinitely little, but so as not to 
: adc’ dyd dez dz! 
feet F, FY, fe. aia 8-5 ge? gs Mes gpa? gia, Bens 
er words, trogurd these quantities as constant in taking the variation of 
be sn. of the. equations. Hence I have ea ant 
