Solution of a Problem in Fluxions. 71 
of the motion changes by insensible degrees, and that there is no fric- 
tion; also the law of foree is supposed to be the same in all these 
cases. (Prin. b. I. sec. viii. prop. 40.) I will now show the use of 
the different forms of F, as given in(L), bya few examples. By 
Cd 
the form F=— e 
p*dr 
is as >, at different points 
of the described curve. _— Fluxions, art. 206.) Also by 
Gis, ¥s Ve 
F= PRR’ Fisas oR R a RQ’ (Prin. b. L. sec. ii. prop. 6. 
V2dp Va 
cor’s 2, 3.) The form a od P= ~ Sives vive 
dr 
; Ve (3); (Vince’s Fluxions, art..208) and F=5 = 5p tives 
42 
R ae 
=35 (4); (Vince’s do. art. 209.) ‘The form Pee an te 
furnishes.a very simple solution of prob. 2. prop. 7. sec. ii. b. I. Prin, 
VP 
(See Newton’s figure.) For R=” r=SP, sin. .b=sin. SPY= 
vr 
=sin, PAV (Euc. 3. 32.) =v’ Substitute these values and reject 
the invariable quantities 2C*, AV?, and there results F as gp; — my 
the same that Newton has found. The same form does also enable 
me to demonstrate very easily the second and third corollaries te the 
same proposition. (See Newton’s fig.) I shall suppose F' to denote 
the force to the centre S; Fs the force to the centre R. Then I 
ae yr (C", eh ip 
ly for S.) Hence I have 
: Sr eer ain. * x 
“7? R sin. ap ° * 7?R’ sin. aff ° SES: sin. 2.” 
have the equations teeth FS, Y = 
being the same for R, as C’, Lon R,L 
Pie: 
Q 
but R= R= r= SP, r= RP; p—SPG=t (Euc. = 32.) ; 
and J/= “he spleen of V (Euc. 3. 32); -*- sin. Y=sin. T and 
sin. J/=sin. V . os “ey T= PV (by trig.); substitute these 
values and there results F ; F’::C’? x RP? xSP : C’? xSG* (5) 
= sites oy Ae 
(since 5753 = SG* by the sim. tri’s TPV, SPG;) hence if 
