Fig. 8, 
FLUXIONS. 
spectively equal to the first and second fluxions of the 
enced smh to the two curves, and.so on of the 
ya hn only coincide in the point 
two. curves in 
in which their ordinates are equal ; el theoiatennt 
the different orders of fluxions merely shews, that no 
ee 
Be! 
i 
i 
i 
i 
: 
e 
i 
F 
i 
i 
i 
i 
i 
(Art. 67.) 
84. Asan illustration of this 
origin 
common abscissa. Let AQ =2; and PQ (takenas any 
ordinate of the curve C D) = y, also PQ 
_ordinate of the strai line TP) 
tangent of =t; then 
straight line PT is evidently u=t¢ (a4+7)=—ta+tx; 
hence, as a and are constant, we have, for every point 
in the straight line TP dvZta<; md *’=t. As be- 
dz 
sides the straight line and curve having a common 
point at P, which is expressed by the equation v = y, 
we farther suppose the nature of the contact to be ‘ 
with what we found in art. 67, formula (1), and thence 
the subtangent TQ may be determined, as was there 
. being thus ined, it is i 
, let 
Put AS=a’ ; tangent of angle S=t’; PQ poli 
— es : eek = consider- 
ed stan oninate of PS) = then the equation of the 
line PS isu =¢' (a’ +2), and hence 5 = ’: Now, in 
order that the line PS may between the curve and 
the straight line PT, it ought t i iti 
moe 
dam dz (#tt- 81.): hence we must have t’/=#, that is 
the nt of the angles PTA, and PSA must be 
cdiiedde + ‘Thos the like is aean eaniten oh 
. a \e. to 
the strictest ion of the term. 
85. Let us now consider the contact of a circle and 
427 
any curve ; let the.circle EPF meet the curve CPD in 
the point P, (Fig. 22.) that AB is their com- 
mon axis, and A the origin of the common abscissa F 
AQ =z; then put y= PQ, considered as an ordinate of 
the curve CPD ; and v=PQ, considered as an ordinate 
of the circle EPF. Let H be the centre of the circle ; 
draw the radius HP, and draw HI, HK perpendicular 
to PQ and AB. Putr= PH, p= AK, q = HK, so that 
- p meg are the co-ordinates of the centre of the circle, 
then HI 
= p— wx, and PI= v—q; andsince, from the 
nature of the circle, PH? = PI? + HI ; therefore 
_ (p22)? + (v —q)? =r Q.) 
First let us suppose that the kind of contact is such 
as is indicated by the equality of the first fluxions of 
di d 
the ordinates ; so that 77 = 7 The preceding equa- 
tion, in which p,q and r are to be considered as con- 
stant quantities, andy as a function of 2, gives us 
—2 ars + 2(v—g)dv= 0, and hence 
' v 
dv _p—< 
- pat pare 2.) 
p—*_ HI_ QN_ QN. dv_QN. 
ea g EET Bg ing MO BE 7 
and #2 y= QN; and since by hypothesis, v= y and 
dv dy d : . 
dsm ds therefore Ay = QN. But this expression 
for QN is identical with that given in art. 67, for the 
subnormal of a curve; therefore QN is the subnormal, 
and uently the centres of all circles, which have 
the kind of contact we are considering, are in.a normal 
to the curve at the common point P. 
When a circle has this kind of contact with a curve, 
no other circle of an equal radius, but whose centre is 
bg: of the normal, can Mey son it mnie 7 
or, su ing it ible, let p’ an e co-ordi« 
nates of the centre of thie Weds eck: and wu its ordi-« 
nate to the abscissa x; then, in like manner, as in the 
former circle, we have found " 
dv_p—« p—« 
we Dears Sf ( yy 
in the other circle, we must similarly have 
du p—r 
a= . 
* vi{r—w—y} 
And as upon the esis that this last circle passes 
between the other circle and the curve, we ought to 
ve (art. 81.) therefore 
ed Phage 
ane ON SET | Fee 
v{r——ay fv {r—w—2)'t 
This tion gives , from which it follows, «that 
dimadan teh soabane have their ‘centres at the 
same point, and therefore are identical. 
The kind of contact which we have been consider- 
ing, which is to the contact of a strai. 
line and curve, may be called a contact of the first 
order. 
86. Let us next suppose the curve and circle to have 
a closer degree of contact, so that not only is y=v, an 
dz~ dz’ dz ~ dz** 
From the second equation of last article we find 
Direct 
Method. 
ig. 22. 
