FUNCTION. 
soak sels if D*. y be negative, the point r is 
d p, or the curve is faid to be concave to the 
axis ; if Ds - y be politive, the point p is between m and r, 
and the curve is faid to be convex to the axis. 
dy 
: ai ta ed ida b+ 2ea, 
pS 24 confequently the curve fs convex 
to the axis. 
2 1 
me dy ama 3x” a 
~ Again, let y = (a) , then <2 and —> 
dw m dw 
a oo ul 
or D’. y =a" — ~# 
1, D. . y is negative, and the curve is concave to the axis. . 
d* y 
If 
dx 
of inflexion: es - this cafe the difference between mm p 
‘and mr or p w(Xa)’ + Diy, (Ax)}* + &e. 
now take. Ax fae ae the term D’. A x), exceeds 
8 D'. 
— 2m 
» , confequently if mis 7 
; or D*. y = 0, the curve is faid to have a point 
r=D 
— fince ie “raft term 
n Aw does, the difference pr r 
oes, or, if on one 
it will be concave. 
If ,» D?.y = o, then the curve will not have a 
point of ‘nflexion, for he a i between m p and m 
will be — (P toy. (Av)t + oy. (Ax)? + &e.), and 
pega) its fign will depend folely on the fign of D'. 
and not on the fign of Aw, for (Ax)* is pofitive whether: 
in 7 is pot tive o > 3 and ies) i - 9 =O, 
,= y= and il = 0, (m even), 
ae the curve has a - lafle exio 
It has appeared, that i eenarally the da 
Fi Dep. (Aw)? + DS cy. (Ax)? + &esy 
aw Oy tg Red Dhey. (Any + 
&c.) ; now fince A # may be taken fo fall, that a term, as 
Ds .y Aa, is greater than the fum of all the fucceeding 
— 
aa it Is that the difference between pr and (Ax). 
De. y or es a: = - (Ax)? may be made lefs than any 
affi gnable quantity: this is all that isto be underftood in the 
prefent method, concerning any equality between a certain 
Rate of pr and .y (A x)*; in the method of flu ranges 
or of prime and ultimate ratios, the “ina ratio of p > OF 
the oo of the arc or evanefcent fubtenfe, is made 
sae d x’ ; not indeed abfolutely and nan! 
r.2.dx 
aa panier only . a conta& 
of the-fiekt order, its equa ona —a-— bt = 0, containin 
only two conftant quantities (a, 4) or elements of conta : 
a eurve whofe equation is w = a@ -++ bi ay in- whic h 
bi are three elements of contact, 
its of a contaét of the fecond pee or io e ometrical 
ifi=mumydueo=dy dt tu 
eat can pafs etween 
= ad Js D the fam 
of the curve cok equation 1s y = 
ite courfe, and th 
9% 
Tn like manner, the equation to a circle containing only 
three conftant seri f: the circle adm} 
med than that of the fec 
¢ cles may merit a particular 
examination, from the rank that the theory of ofculation or 
of circles of curvature holds in ¢ ory = facies 
Newton was led by it to his theory of caitel for 
Let circle and curve aie a common n pint P, then: 
fincee PL* + LO? r(PM — LM)? + 
(AN AM)? OF putting PM = nL Moron 
= 4, AN=a4AM= tt (u — 6)? + (a — t) : 
and confequently u = § + V(r — (a — ty’ a siihecoms 
mon pointw = y,f = a y= bt J (? _ (2—.«)° ) 
d 7 
. wo D.y= (a— 2) (Ff - (4 — a))-4; and ince - 
LO:OP::MK:PK 
@—e: res yun Vi+Dy 3: D .. ¥ ¢. 
71+ Dy’, 
re ad 
a-xk= = 
but PL: PO: teat: PK 
yo—bursi yy V1 + (Dy)* 
oes 
4 r+ (Dy)’ 
hence, @ = x + aia and 6 = yom 
pe de yea 
(1 + (D >) — 
r 
and confequently, in terms of ‘y or 
/ r+ ( ? g Yo J § 43° 
the co-or dinates a and 4, by which the centre is determined, . 
are sxPn pene 
he and 6; on which the degree of conta 
ca ine cae ae hm on thefe two.conditions, viz. . 
dy 
and that n= = = sand agreeably:to — 
laid. down, the circle whofe radius.is r, and the po 
ee centre 1s determined by the values of the estas: 
as above, is fuch,. that — it and the curve no - 
le of fam n be 
that vu = y, 
eae circle of the e radius drawn, . having its - 
centre placed differently. 
For. if poffible, i ‘the equation to fuch a circle be 
(o.— 8)? + (a a— tm orv= B+ f(r — (a —2)"),. 
at the commen point v= y, hence,.e.and 8 are to be 
determined from the equations y = 8 + 4/ ‘(4 — (@—-x) ) 
and D x) (+ — (% — x)? ae 4, which a 
tions are exaflly fimilar to thofe from which a and} w 
determined : confequently « and deduced, will be expreffed. 
by the fame quantities as 2 and } are, or ates w circle will ° 
— identical, or coincide, with the form 
e the conclufions deduced as above om not depend on 
the anes r 
for a and bs 
petshals 
D and from the other (y — B) 
( 1+ (D. ys equate the two values, and there refults 
@—e¢= (y— 4).D.y, orb = y 4+ —— an equa. 
tion to.a right line in which are placed the centres of all the 
circles 
