SS Sere 
Fig. 31. 
FLUXIONS: 
Sie, neal te fecen at nearly = 1, Our.li- 
us torefer the teader. to the fol- 
ay er Se 
he Gp sy — Comm. Pet toma Xviii. te 
Mem. de UAcad. 1786p: 620; and 
Exercises de Calc. Integ. Ivory, Edin. Frank ‘Vol. IV. 
Wallace, Edin. Trans. Vol. V,. Woodhouse, Piil. 
7 ioe eats os hype found exact 
length of a ic arc may be 
ithe same mane, omits equation cy = Va —1) 
158. There is a very ic arcs, 
property of 
171 Op. T. IL. p. 
sir) which has igen iby fe aes in the 
It be easily 
fl gn ar 7 rin ei of ce 
a at in 
(Fig. 31.) Draw C from the centre icular to 
EN ap A tantra 
con to wn 
tno D: also, ee nen dicular to the axis, 
As et, let CA We =c, the eccentrici- 
Tae CE =p, DE=t, Hp .snale 
AcE Cece ea the elliptic are AD = 
Because HK*= c# (1—CK?) und HK = CH 5 <cos. % 
also CK = CH x sin. @, therefore CH? x cos29= 
(1 CHP x sin’ ¢), and hence 
CH? = con aint? , and putting 1 — sin.? 9 for 
cos,? 9, and e* for 1 —~ c* in the denominator, 
c 
cH= J/U—é an. %)? 
1 Poe 6) CH x curve, (Conic Sec- 
20,) CH x CE=AC x CB=c, 
phaiebe W(1L —e? sin. 9), 
“fire gas 
And since in curve 
ofthis kind, (at. 77.) d (+ 1) = pd@, { 
Pye, eek alah eat hie 
COs tah hence ae ee ee oe es 
eee ad : Let usnow sup- 
« = sin. Q, then it follows that dz = do cos. 9 = 
Gop — 2*); we have also 1 —e? 221 eee e 
And en b 
TION 
Sesh bea eeiiconattg (2.) 
pe formule (1.) and (2.) it immediately follows 
d (241) =dz, and? +¢#=24 ¢. 
ic ant, draw D a tangent to 
451 
assignable straight lin 
ce whch hsb ei ‘been asigable sight Tine 
6 ere the hyperbola, 
ag much to. discuss 
On this subject, con- 
ig Te Euler, Nov. 
T. vi. vii. aie Act. Acad. Pet. 1778, Pars, 
at Fag 
iscell, 
Hee phd ekg Ok 
The T. iv and’ "Theorie des Fonct. Ana be 
pena 
“edit. 
Legendre, Exer. de Cal. Integ. and Mem. de l Acad,1786, 
ony a ao Memoirs, vol. i, ; Lacroix, Traité 
‘al. Diff: et ante, vol. ii. ‘Also; ‘Wallace, Edin, 
Trans. “voles . Brinley, Trish Trans. ; Woodhouse, 
Lond. Trans, ; ‘Ivory; in Leybourn’s Mathematical Repo= 
Ba Part ii. p. 9- 
x, 5. If we employ the same construction and the 
same notation in the hyperbo yperbola, (Fig. 32. 
lips that if DE boa tngene st By 
=e, the 
CEa 
t, and if we 
bolie ‘arc 
ar CE=p, 
d exactly as in “he 
kip: pat way ?), and (art. 77) 
d(e =d9/(1 —e? sin.? ?) en (3.) 
In the ellipse e¢ is less than unity, but here is great« 
oe hay i sree Sie eiemanetance MK an essential dif 
in the two fluxions: the one form is, however, 
neible to the other, as, was, first shewn by, Landen in, 
Phil. Trans, 1775. In either case, the fluent. may 
ml by re -/ (1 —e? sin.? Q) into a series, 
term by d@, and taking the 
(Fig. 23,) let F be 
E intone at-P, and FE.a 
ar to Pip op nti =a; 
perpendicul 
the angle AFE = 9, the vse E= p, the.tan. 
PE=t, the areP A‘ 5} we have found (at 89 Bx. 1) 
that in the parabola p =,——., hence © RS Mit, 
Ee 6. nar in. the. 
Pt yy eee 
foc B 
ado 
Teos, 9? Andy by arte 146. 
aod 
iota 3 [ioe 1, fein. (ge eae b 
This fluent wants no constant ‘quantity, because when 
? =0, both sides yanish as they ought. ° 
From this we assign parabolic ‘ares, 
which should be to each other thie givep Moyea 
number to another, which was first done by John'Ber- 
gat Ce ese a 
, ; 82. g 
thé parabol ee relation to Fagnani’s theorems, 
Ex. 7. Let the curve 
33,) of which DC is the axis, DH’ a circle d 
on the axis, of which the radius = 1, PG an ordinate 
to the axis, which meets ‘arcle ih H. Put DG=z, 
GP =y, cire. are. DH =», cycloidal DP=z. By the 
nature of the curve (Evieycrom) 21 — cos. v, y= 
nv +-sin. v, where n is agiven number, which in the 
common cycloid is unity. ence Sd sift 6, ag= 
et Nba °) ad v/ (1 nt 2 
z= of (da? + v +2n cos, v). 
Instead 2 ae v, ie Toa taape, ‘and put e for 
a oe and we have ; 
Tet 
a 
as in the Fig. 32. 
Fig. 23. 
a cycloid of any kind, (Fig. pig, 39. 
