FLUXIONS. 
50 i 
*•' ^jcZaZ-lf-a 2 —c 2 X4 fl *— 4.x 2 
za ax — x 2 
and the radius of cur- wz 
vature 
■030 - 
a 2 c 2 +a 2 — c 2 X4-ax —4* 
za A c 
J2I2 
a 2 c 2 \2 
, or -is ; agreeable to the definition of a circle. 
Whence (making z conftant) x 
i 2 y,a—z 
af zaz — z 2 
which we get RO, or AO ^=2^-^ 2=4/2 az —z 2 ,andrO, 
or AS ( — 
(=f-,=- 
— which, when z—a, or ROH 
coincides with BH, become AOH (BH) 222/7, and CH 
(AG) — \a. Hence, becaufe it appears that, JvH] 2 (a 2 ) 
: AO 2 (2 az—z 2 ) :: AG (i<z) : AS ( -—— ) it fol¬ 
lows that the evolute AOH is alfo a cycloid equal, and 
fimilar, to the involute ARB. 
If the evolute had been given, or fuppofed, a cycloid, 
and the involute required, the procefs would have been, 
more Ample, as follows: Let AH (2AG) —a, AO 
(=RO) —z, AS—x, SO=j', BR=c/, B hz=zw } Rr—v } 
Rtzzw, See. Then it will be, 
y : z (:: Om : OR) :: Rt (w) ; R 
y :: z (RO) : O m — 
z y 
z x x :: z (RO) : Rot 222 —, 
Whence we have v =-^£, Rn (Rot—AS) =2 — — x, and 
y ^ 
z y 
An (OS—O m) 2=7—; which expreflions anfwer to 
any curve whatever. 
But, in the cafe above propofed, AH 2 (a 2 ) : AO 2 
z ^ zz 
(z 2 ) :: AG ( 4 -a) : AS ( x ) ; therefore xz =.—, x —— and 
2 a a 
yi^fz 2 —^.2^ — ^y.. a -L; andconfequently R?;^L^— x) 
:-ia—w (or CB—-B/r) : whence alfo 
Vol. VII. No. 444. 
3 — 2 2 , • / C IV Z 
-, and v { —r 
2a V y 
> 
— : there- 
whicli 
when the diameters- a and c are equal, or the ellipfis 
degenerates to a circle, will be every where equal to 
Ex. 4. To find the radius of curvature, and the evo¬ 
lute of the common cycloid. Let ARB be the given 
curve, and AOH its evolute; alfo let R/: and OS be pa¬ 
rallel to AC, and eO and Rm perpendicular to AC ; 
|/ a 2 - Z 2 4 / 2(201 
fore it will be v : zv (:: a : 4/2 au>) :: 4 /\a : -4/; that 
is, as Rr : Rt :: 4/BC : 4/BA : which is a known pro. 
perty of the cycloid. 
Hitherto regard has been had to curves where the ordi¬ 
nates are parallel to each other: but when the ordinates 
are all referred to a given point, as in fpirals, &c. other 
theorems will become necelfary; and may be thus de¬ 
rived. 
79. Let ARB be the propofed curve, P the point, or 
center, to which its ordinates are referred, NOL the evo¬ 
lute, and RO the ray of curvature at R : 
moreover, let PH be perpendicular to 
RO ; and, fuppofing the ordinate PR (jy) 
to become variable by the motion of the 
point R along the curve, let the fluxions 
of AR and PH ( p ), exprefling the celeri¬ 
ties of the points R and H in directions 
perpendicular to RO, be denoted by 
z and p refpeCtively. 
Therefore, the celerities of any two points, in a right- 
line revolving about a center, being as the diflances from 
that center, it follows that p : z :: OH : OR ; whence by 
divifion (putting RH=:t-) we have z—p •. z :i v (RH) : 
VZ Vpi, 
but pzzzyy ; and there- 
RO =2 — 
from 
fore 
~v 2 
vyy 
z—p 
RO =2 
pz—pp 
v yy 
(and 
yy 
• ; which, becaufe y 
yy—pp ’ y 
therefore yy — pp—vv) 
2 —p 2 
will alfo be 
vv nj 
The fame otherwife. —Let SRD be a circle deferibed 
about the point O, as a center, and fuppofe the diftance 
PR to be variable by the motion of 
the point R along the arch of the cir¬ 
cle (inftead of the curve) : then, draw¬ 
ing OP, and putting ORz:r, PR^rj', 
&c. as before, we ftiall get OP 2 (OR 2 
+ PR 2 — 2OR x RH) =22 r 2 -j- y 2 . 
—2 rv ; which (as well as r) being SOD 
a conftant quantity, its fluxion 2 .yy —2 rv mull be 
yy 
equal to nothing; and therefore r— the very 
fame as above. Nor is it of any confequence whether_y 
and v be here looked upon as refpeCting the circle, or 
the curve; fince, at R, they muft be the fame in both 
cafes ; othewife the curvature could not be the fame, 
(Art. 74.) Now from the value of RO thus found, 
which (corrected when necelfary) will alfo exprefs the 
length of the arch NO of the evolute, (Art. 74.) The 
ordinate PO and the tangent OH of the evolute may be 
eafily deduced. For OH (RO—RH) — v — and 
PO (=4/OH 2 -|-PH 2 ) — PVP ; '<-v 2 ; whence the nature 
v 
of the evolute is known. 
80. Ex. 1. Let the given curve AR be the logarith¬ 
mic fpiral, whofe nature is fuch, that the angle PRQ 
(or RPH) which the ordinate makes with the curve is 
every where the fame. 
Then (denoting tire fine of that angle by b, and the 
radius of the tables by a) we have RH (v) — and 
a 
6 M therefore 
