Fig. 21. 
Limit 4 Sdeyxiimit 5 
the fluxional ratios instead of 
and regarding 
On the Contact of Curves, and Circle of Curvature. 
81. Let two curves, CD, ed have the same axis AB, 
Ce renee eee . of their 
co-ordinates ; let y= PQ’, and v=P"Q be ir respec- 
tive to the common abscissa 
z=AQ, and let their equations be y=f(x), and v=F(z). 
Let us also , that when x est+h, y 
comes y’, and » becomes v’: then, by Taylor's theorem, 
y yah dty ke 
du he 
taint! Ship See rik + &e. 
ieoureree i Hae sebnrgsrto- bare 9-comnarn. Belial, 
nyo on ; the nature of their 
contact at that point is such, Godt ae tte deviate 
the same common point, can pass between them, unless 
peli vs me cor fiver that is, supposing w to be the 
curve, corresponding to the com- 
a aaa 
so 
&u he 
Waugh + SES 4 80 
this curve, wie 
t dv 
wo, unless 9 = 2 4°, For let us suppose it pos- 
sible ; bovine ga, prac hh ae 0 ex 
mee (EG) A+ (S292) 5 ae 
and by hypothesis, y= v, also 4% = $°, therefore, 
(ty _ doy ¢hy Boy h3 4 
ore (ae rr 2*(73 aa) e t+ * 
expression for the excess of P’ 5 
di veainas the fin Grve shove DQ, the ordinate 
of the second, In like ‘manner, the difference of the 3 
— + ke, 
Se eee ee 
¥ —w Hye gata + &e. 
or, because y = u, 
smen(E—$) 4 (GEE) Ea 
apa tp to pass be- 
FLU XIONS.. 
(Ger te) F+ (a-3 &e 
re oe a 
and hence chy 
(G- Ps(S—$2)8 ou 
dy ad Ah 
> (i-z) + (75-Ts) a+ & 
This ought to be true Sonsevenprveioe of h Widlbhets 
than But h to the first 
of these <roeupesatieetaas Rapiuiions teat 
assignable, because h enters into all ts torme, while the 
second approaches to the limit =a “Now this. 
— 7, = © that is, 
,» for Wed ; 
3 (25 2) etem(Sh Baa 
which “is evidently” possible. 
‘is tly Hence between the 
courses of the two curves, which have a common ordi- 
nate y=v, and which have also = 52, no other earve 
can pass ; unless the Muxion of ts nde x be equal: 
to the fluxion of y or of v; co that 44 = 4. 
2. Again ; if in the two curves, whose equations are 
y= hae v=F (x), and which have a common ordi- 
nate y=, we have also 4¥= 9, ana Y= © ; then, 
no third curve, of which pth: wiretap met x), 
and which has a common ordinate with 
can pass between them, ales a the sae ime 
dy du aty— 
as ae ee Ga = For if it were possible, we 
ould, as in last article, have y/—o' > y’—w, or, 
jecting the quantities that toatl cach othe) Lae 
at doy \ h3 + Ae & 
—as)et+( cay an m 
du uy “a i. 
Te) Sea act 
Now this cannot hold true, unless 22 — — 7 =0,and 
also 4 S% 0; for, were this not the case, k 
ponsrngtrs Retire de, pedicels 
amount of the first of these two be 
iy 
He 
ae 
iF 
te] 
| 
BEE 
25° 
in 
Th 
oF 
Yawk the ‘rine cliteetin, bene 
tae Nye povee of A. Er owerte, 
i u 
dye and ——~ iam = 7,2? then, that the third curve 
re Mra seein se oe | requisite that 
d3v 
da Za) +e: 2 (Topiges ailalle 
which is certainly possible. : 
88, In general, if there becuny: carve 
» Seen re ee ee it, 
which requires that their re es 9 
the same abscissa should be equal ; ‘then, ff the first 
EE ——. 
= SS Regrets te 
