FUNCTION, 
— ore touch = ke that is, that have with ita y + Ayis 7 y. or and < 
£ the firft 
_ ai of nin Psi ormal has beem found = y. D. y 
and the — of the an which the normal a with 
eqn 
the axis is » fuppofe-now PK to be the normal, 
D- 
then the equation to it, confidered as a right line is (fince 
PM = ota MN x tangent 2 PKM + 
w—a 
ON) y= —s 
the fame’ ase as has been ot deduced for the right line 
in which the centres of all the circles that touch ve curve 
are placed : nee; “this right a coincides with the nor- 
ae on is confequently perpendicular to the canseat of 
2 as, confequently, 6 = y + 
poste it now were required, from all the circles which 
touch the curves that is, which fatisfy thefe conditions, wiz. 
y=, 2 = a to feleét that which has with the given 
¥ 
eurve a contact, fuch, that between it and the curve no other 
eircle can be drawn, o or whieh, analytically {peaking, fatisfies 
ay. da 
the condition ia = ha : now, by what has preceded, 
and 5 = y — 
- 
=e+ ? sr: 
. we a oF) cs + (D.y)") 
whem sngis(2)az 
= (P— aa) pao 
Pe oe — (D.)*)t 
see 
Da)! 
1+ (D.)) 
7 D. 3 dx +d 
and confequently, r= (7 ( nr) = eri 7 
D. as (@ dyi(dart+td 
anda=x— — 2 st ‘ Dore 2k = ‘A 
+ 2 ad d , 
andb = y+ “7 Os, =y+ i. 
By thefe sis ae peairtine ac the co-ordinates of the 
radius of that circle which is fuch, 
ve no other circle can pafs; for, 
i pafs between de 
eurve and the former circle, let its cade be v-== 6 
af (¢? — (a —¢)* *), then the three elements of contaét, to 
_ By ba ey are to be determined from the three equations 
PvE GP hDua es 
ia _ osu j= Fae a 
sircle will econ identical: with the form 
ya bake to invefti te an fe eepretion from 
reas of cies may be found. anppe ordioae 
ordinate wy the curve, aie a an elo w the ordinate 
isy+Ay=y- + Dy. Axc+D vy. (Oz) + ke. 
and the incremental: area between . the . ordinates Jy. and 
(y + Ay). Aw, the ure 
ncreafin _Aa, and > (y + Ay), the 
ordinates "timinithin ng; let now the area be Aarne by 
Ie a funtion of ne abfciffa, then f. (x ve 7 _ 
“a, OF ace w is between the limits (y Ay). and 
Vi oes ae et ie be Tee rine 
(y+Ay). , ower {mall A xis, or putt ig 
for A uy = hee seri feries, (Diu —y). Ax 
DY .u(a 74. &c. oe hae or - Diy. (Ax) 
7 Dy. a we)? + &e. and aeeametly (D.#— yy) 
+D iy, Aw + &c. mut be 2 D.y. Aw +D>.y 
eke 
(A v)* + — however {mall 4 x is, which is ev vidently i im-- 
poffible, exc .u — y = 0; for the oe of the 
two hee a! being (D.u ~ y) + (D -u—D.y 
-Aaxt+ (Dox —D*. yy (A ae ge. it is evident 
tA: be taken fuch, that (D.u — y) thall be- 
ne tea ne fum of all ae eats terms ; hence 
et 
ef 
D.zu —yors = =yordu= yd; and confequently 
x 
d7 2 ory = d“ (yd). 
Hence to find the area is merely an analytical laa a 
ordinate ea oe a funtion of the abfciffa; wha 
to he dee isin os of the integral of a 
iven ileental exprefion 
Ify=a+ba® +e 
fs urn da batda +4- 
a cnt 
ex" dx,andu=axe 4 —— + ++ a, a being anar- - 
ae ro r- 
bitrary quantity, which in fj afe y be determined. 
The differential . euiehion from which ae folidity of a’ 
body is to be calculatéd, may be 
Let f« be a fection of the folid, ne perpendicular to 
the axis; let s, a-function of the-abfciffa, reprefent*t 
oe .then As : is contained between oa limits f (# +- A a) 
ee --Q x, confequently As— fe. Axis <. 
(Fla + Aa) — fa). amor (Ds nee Te 
Ax)? + kcis< Dfae. (Ax) + D fx. (Aw)? + &e- 
= pees (D.s—fx) + D¥s. Aw + &. < D- 
fae. Ant D> fe. (Ax)? + &e. “accu {mall Aw be- 
rede which is impoffible, except D..s— fu = o, rae 
may be taken fo {mall,. that (Dfz — D eS 
+ a. (2 2 + a’. awe e A &e. (putting a, a' &e. for : 
v d witl hi(a v)'y (A x)’ 7X c.) 
is lela than D .s — f.¥, and a fortiori Jefs for all. fraaller- 
values of A x3. hence D.s =f x, or = 
dx 
= fax. if the folid.can be concei ived- to be generated’ 
by the lees of a figure round - axis, the fection, the - 
area of which is reprefented by fay a cece, confequently. 
reap yas ra ye a ttt59 ee Se = py, and: 
thence ade 
The differential exprefon 1 ext to be deduced | is that on: 
=f, ords. 
i, the limits . 
within. 
