FroP. LXX'VII. —To determine the ratio of the velocities of 
todies revolving in different orbits, about the fame or different 
tenters ; the orbits themfelves , and the forces whereby they are de¬ 
fer ibed, being given. 
155. Let AQH be an orbit, deferibed about the center 
of force C, and let the force itfelf at the principal vertex 
A be denoted by F ; alfo let r (land for the femiparame. 
ter, or the ray of curvature at the vertex, and let CP be 
perpendicular to the tangent Q.P. 
Then, the celerity at A being, always, as frY (by Art. 
148.) we have CP: CA :: frY (the 
, . , CAXt/rF 
velocity at A) to 
FLUXIONS. ass 
X P) the required-time of one revolution 
to 
f AO -2 
the 
CP 
locity at Which anfwers in all 
cafes, let the values of AC, r, and F, be 
V CT i 
when the orbit is an ellipfis; that is, when n s is lefs than 
id 
2 : for, if n 2 be —2, the curve (as its axis-- becomes 
2— n 2 
infinite) will degenerate to a parabola ; and, if n 2 be greater 
than 2, the axis being negative, it is then an hyperbola , 
• . id 
whofe two principal diameters are equal to —— and 
imnd 
f AC O B 
what they will. 
Cor. 1. If the centripetal force be as the fquare of the 
diftance inverfely, or F be expounded by the velo- 
AC / ■■ 1/r 
city at (Twill become __ X J— 1 —, or ^5 : Whence 
CP > AC 2 
the velocities, in different orbits, about the fame center, 
are in the fubduplicate ratio of the parameters, and the 
inverfe ratio of the perpendiculars from the center of 
force to the tangents, conjundtly. 
Cor. 2. Hence, if the celebrity at QJm denoted by 
Q.7, and Cq be drawn; then, Q cq being as > if follows 
that f r is as CPx Q7, or as the triangle QC q: therefore 
the areas deferibed about a common center of force in a 
given time, are in the fubduplicate ratio of the para¬ 
meters. 
Cor. 3. Laftly, fince the area of the curve ACTHB, 
&c. when an ellipfe, is known to be as (AOxOD) AO 
X f rx AO (fuppofing O to be the center) if the fame 
be applied to 4/r, expreffing the area deferibed in a given 
part of time (by the laft Carol .) we fhall thence have AO 
X V AO, or AO 2 for the meafure of the time of one 
whole revolution. From whence it appears, that the pe¬ 
riodic times, let the fpecies of the ellipfes be what they 
will, are in the fefquiplicate ratio of their principal axes. 
Prop. LXXVIII ,—XThe centripetalforce, tending to a given 
point C, being as theJquare of the difances reciprocally, and 
the direElion and velocity of a body at any point Of ring given ; 
to determine the path in which the body moves, and the periodic 
time , in cafe it returns. 
X. 156. It is evident from Prop, lxxiv. 
q F.x. 2, 3, that the trajectory AQB is a 
\N. conic fedlion ; whereof the point C isone 
^ ' of the foci. Let F be the other focus, 
j '.and upon the tangent PQjC let fall the 
perpendiculars CP and FK, and let CQ__, 
and F(T_be drawn : alfo put the femi-tranlverfe axis AO 
=za, the given focal diftance CQ =cd, and the fine of the 
angle of direction CQP (to the radius 1) —m ; and let 
the given velocity at Q^be to that whereby tire bo'dy 
might revolve in a circle about the center C, at that dif- 
tance, in any given ratio of n to unity : then it will be 
n : x :: FQ^: AO 2 (by Art. 153.) therefore n 2 : 1 2 : F() 
d 
(2 a — d) : AO (a) ; whence AO (a) is given — —. 
Moreover, fince CP=»zx CQ, and FK^bxFQ, we have 
w 2 « 2 2 
QD 2 (=eCPxFK=ffl 5 x CQx FQ=~— ff’■> whence the 
femi-conjugate axis (OD) is given likewife. 
Laftly, it will be (by Art. 155.) as CT" 2 : ACf 2 :: (P) 
the periodic time in any given circle, whofe radius is CT, 
f n 2 —2 
Cor. Seeing neither the value of AO, nor that of 
the periodic time, & affected with m, it follows that the 
principal axis, and the periodic time, will remain invari¬ 
able, if the velocity at Q^be the fame, let the direction 
at that point be what it will. 
The fame folution may likewife be brought out, from 
Art. 155. by firft finding the principal parameter: for, it is 
evident that the area deferibed by the body about the 
center C, in any given time, is to the area deferibed, in 
the fame time, by another body revolving in a circle at 
the diftance CQ, as mn to unity : hence, it will be i 2 : 
tn 2 n 2 :: d : (m 2 n 2 d) the femi-parameter : from which 
(proceeding as above) we get aXm 2 n 2 d (= 0 D 2 ) —m 2 y. 
2 ad — d 2 ; and confequently a — -——, the fame as be¬ 
fore. 
Prop. LXXIX.— The centripetal force being as any power 
(n) of the difance , and the direction and velocity oj a body at any 
point A being given, to determine the orbit or traie&ory. 
157. From the center of force C, to any point B in the 
required trajectory ABD, let CB be drawn; join C, A, 
and let A b be the given direction of 
the body at the point A, and Cb per¬ 
pendicular thereto ; alfo let the velo- 
-tp city at A be to that whereby a body 
might deferibe a circle AEF, about 
the center C, in any given ratio of p 
to unity ; putting CA=«, and CB-w 
then, becaufe this laft velocity (the 
centripetal force being as x" or a") is 
H- 1 
r rightly defined by a 2 , the velocity 
D /F of the body at A.will be truly exprelkd- 
71 4 “ * 
by pa 2 . 
Moreover it has been proved, that if the celerity, at any 
given diftance a from the center, be denoted by c, the cele¬ 
rity at any other diftance x will be truly reprefented by 
n 4 ' 1 
J c2 t . wh ence, paT 2 ” bein S fttbrtituted 
n -\-1 
I 2 2X 
fore, we have Jp 2 + —7—Xa’ i+1 
riA-i 
nff 1 
for the cele¬ 
rity at B. 
Blit now, to determine the curve itfelt from hence, let 
BP be a tangent to it at B, and CP perpendicular to BP ; 
alfo let CB, produced, meet the periphery ot the circle 
in E ; putting the arch AE ~z, the ioreiaid velocity at 
B (to fhorten the operation) =2 v, andC bz=sb: then it will 
be v : c (the velocity at A) :: b : CP —-: whence BP 
V x~ v 2 ■ ■ o 2 '. 2 
CP 
( = 3/CB 2 —CP 2 ) = 
/ I_ 
Moreover we have, as CB: CP :: v ■: ( - x v) the • 
VCB 
velocity of the body, at B-in a direction perpendicular to 
CE. : . 
