FUNCTION. 
D.y.dx4P iy, (Aa)? + D3. y. (Ax + &e. 
D.u.Ax Pe ue. (A x? + D3 .u (Az)? + &e. 
D.v.Ax+ ee v.(A x)? + &e. 
it follows that (D.y — D.») Agwt ofan eee! rae v) » 
(A‘x)' &c. mut always be lefs than (Psy — Ds a). 
{Ax + (D-, “I — a). (Ax) + &e. aa the — firtt 
- terms in the difference A y —An,towit, (D. —D.un),; 
(PF. y — 2 , by hypothefis = ©) ; but this is = 
impofbble, except both D.y—D.4,andD*.y — 
n be taken of fuch a magnitude stat 
@. i _ D.») “thall be greater than (D2 yn WAG 
—_ (Gar yr Ds =) < (Aay — &c.or (in eye au th, “y 
=. v) fhall be eee than 
and general y>if two curves whofe equations are 
e thefe co oe? pe atthe me point of the abiciffa, 
2 eee r,D. 
=D. ys u, D3 
&e. Dp . 7 = pe. then een "thefe 
wx), dra 
J = ly Uy 
two curves no other curve whofe agai isv= } wn 
through the common poe = an pais, ex .v=D.¥%, 
a = D:. pa and D’-'. uv 
n—! 
; for, 1 fick a curve aa pafs “between the 
pees a ie iso other curves, then A y — Av mutt .be 
lefs than Ay — Aw, or lefs than mae a Da . oe (A x)” 
+ on ym ds “t) % (Ax)"** + &c. ‘owes {mall 
Ox is; but this i is impoffible, fince the term s of the feries 
re ne Sy—Ovmu involve po 
Ns D*.v= D: e J D:. — 
» XC. D = DD". y; and if thefe differen. 
tials are not ak then aiding by a certain ae of Ax 
Ay - ee iat: be on er AS 7 »ifina fe 
res, asa a a‘. Pas A .a@ can ge made to 
exceed the fum of i ie ine which evidently can 
Ls done by taking A » of a proper fmallnefs. 
iefe are the principles on which the contaét of curves 
; the contact admits of diffe rent degrees ; and 
arranging fe contats according to different orders, there 
mit wi given curve only o contacts 
a certain order; for initance, a right line is oie only 
of a.contaét of the-firlt order ; a circle is only capable of a 
contaa& of the — order ; a curve, whole’ equation is 
xv, is not capable of a conta 
higher than — cf the third order; acurve, whofe equation 
Lat + cui +da' +e x’, is not capable of a con- 
r ce that of the fou 
that admitting thefe different degrees or orders. of contact, 
i in number of conftant quantities, which 
a contac of the firlt 
om 
p 
o 
~~) 
Mt 
a 
qe 
=] 
5 
fe) 
ct 
mth order — + yee on ree err 
Asa mple, let u = 
‘toa ftraight cae ; then fince - is to Ege throu 
: a+b 
‘the curve, u-= y and #. = 4, 
dy _ 
dz 
ugh a point of 
* D-yo 
=b,anda=y—x.- b= Jee D. “Js confequently 
D.y'+ 
the abfciffa # being ccna Cae 
. The line determined by this equation is commonly called 
a tangent ; and, agreeably to the a of contacts laid 
the cleats to the ftraight line is x = y — 
D.iy-ty 
wers of A x lefs haa 
3, which is thie equation ? 
Aw fuch, that 
all the fucceeding termé, then p r will be negative or pofi. 
do its geometrical property confifts in this, that no other 
feraight Hine can pafs between it and the curve whofe equa 
tion. is for, if poflible, fuppofe a ftraight ine 
whofe equation isv = a + Bf, to pafs between the curve 
dy 
dz 
=y—w#.D. A and 
and line, then fince wv = t= key at Bx, o% 
rD, A, and a —Bx 
- ee to ae fiat ee isv = a“. 
yt, which is the fame equation as was de acer for the 
former itraight line, confequently this sai ftraight line 
coincides or becomes identical with the for 
In the eve ux a 
t, AB (PL e VEL. Ath 
fig. 6.) = 
+ dé 
ae b = tangent of angle 4 
(radius 1) but 4 = % — , = 
and this is the value of the line TM called the 
, and confequently T M = 
pda 
ay 
fubtangent 5 oo in ‘a given equation to a curve, find 
the value of 2 
7 —, andmake T M = ye "5° jon T P, 
and the line T Pisa tangent to the curve, 7 has the geomie~ 
trical property before {pecifie 
dy 
dx 
confequently the fubtangent 1 = ye 
Example—Let y = Gx = (2 rx — «)%, then 
ro 
re 
(2ra— wv) 
(2re— Ue x 
r—x 
Agi o the equation be «'~— 3axy + 9° = 0, then 
7d gavdy—3ayda + 3y' dy = 0, and 
Kgxou— 
‘da 3a%—3y ax = 7 
1 men —1 neg a y ~~ 
confequently rae reer and the fub 
tangent us = = ea ITF 
a —ay 
Let a: “K P be drawn pend to the tangent 
TP, then it equals y “(1 + (D-y)’), ory ¥ 
< EY (2), for TM: PM: PM: MK3y.9" 
_ 
d 
gry SL = MK; PKs y (32) i. Pe 
v/ 1+ (Dy)’, becaufe P K is fecant ‘othe ae K, 
dy 
whofe tangent is 
8 dx 
and rad. 13; and confequently, K M 
d 
the line K M the fubnormal. | 
Draw the line mp parallel to MP, thea : ve = 
tx tan ZPTM=y + 4 
i=y 
Z, = Ory Dy; the aad K P is called the normal, 
MP + 
Aw=yt+ 
f $2. Aw; ory + D.y. Ax, butmr = y + D.y. Ax 
as D. yr (A i = 8c. -confequently, the difference or 
pr= —(@, ys (A ay + Ds “Je (A +) + &e.), take 
the firft term fhall be greater than the fum of 
tive, as )? . y is pofitive or negative, for’ (A - is always 
.3Me2 ofitive 
