FUNCTION. 
within which the are is contained being determined, the dif- 
ferential expreffion by which its length is to’ be calculated 
may be obtained thus: 
perpendicularto Z P»; let M'r,M Q, be 
tangents, “then the arc M M’ 7 chord M M’is 7 M'r 
(fuppofing oe r — : si ae axis to the left of MP’; 
again, arc + #Q, but arc M’Q is > :Q 
(for fame ron as arc M MC >M'r); confequently arc 
MM is < Mz >M'r now if PP’ be Ax 
Ax ve + (D 9) F and calling P! M’,y', Mir =Ax 
a/(1 + (D-y)), therefore arc MM’, or fuppofing x a 
funGtion of « to reprefent the arc, Az is (Ax V(t + 
(D.y)'),> Ax J(1 + (D.¥)*), and confequently 
Akl (x + (D 9) — Az ought to be 
<Ae / (1 + (D.y*) — Ag AG: +(D. 
but Az = Dizsi.Agk + De, , (A x)” + &e. 
y=y+D.y. Avy Dey, (Ax)? + &e. 
yy) 
* confe- 
quently D . <y' == Ds y+D. y- Sede ~ (Ax) ¢ 
+ &e. eC +(D.yxY) = VE + ea +2. 
D 
Diy(Dr.y de tard. (Aa) + &e) = V(t + 
(D. yy’) +a. Nut (A x)? 4+ &e. putting “a, a! 
&c. for the co-efficients of i x. (Aax)*, &e. hence 
(va + (D.9)*) — Diz) Ae —P oz. (Ae) — 
Ds .%. (Ax)? — &c. muft be <e. (Ax)? + a'.(Ax)i+ 
“(vo -yY) —D.z) — Doe. As — Be. < 
-+ a. (A wx)* + &c. howev er {mall & s, but, as it 
he “been thew i in preceding cafes, oe is impofiible, except 
{1-4+ (D.y)*) — D.2 = 0; confequently, fince the 
Gifference between the arc and one of its limits muft be lefs 
than the difference between the ‘limits, V(% + (D.9) y= 
D. Senne Per oer 2= /(1+(D. ee 
ds Zz 
or — =. avd (: | 32) ) 
_ ard2 = a 
Ex. 1 Let eye then dy= ve xt.dax, 
and dy = ake a eg aga Se (4 4 i 
2) confequently, dz = (da? + dy" = PrAce 
44at+9 2) 
27 fa 
an arbitrary quantity called the correction ; if z = A, when 
zor, thn A = e+. 91°) i 
an eae 
+ 9 «), and integrating 2 = + a @ being 
+ a, confequently, = 
A (4a+or): 
, 27 a 
: ty a ae 
7 Dy’. ; Aw) 
ieee mae petit 
dys ee da + 2 
oe » confequently, d x ry, ae = a = (by expanfion), | 
dw (1-?#—Dr4.x 4D: 1.2 — &c.) (in whch 
D 174, D: 1~8, &c. are ufed as fymbols, ia — 
Di iz D: 
tively —4, 3, &c) ands =*- + x74 = 
x 3 x . 
+ &c. + aor = + aes 508 4+ &e. + a, (xan arbi- 
trary conftant quantity). 
Let it now be required to inveftigate the dif ferential ex- 
preffion from which the furface of a folid gh) be’ he duc 
It has been fhewn that the arc M M'is 2 > Mir: 
now if the figure be conceived to revolve round ¥1 P, then 
T, P M’ deferibe circles, Mz, M'r conical furfaces, and 
MM! aconoidal furface, Wiel w at be contained between the 
conical furfaces generated by M t, M'r ; nowthe conical furface 
belonging to Mt =4(2p.Pe+2p. ) Mt(p=3 
14159 + &c.) = p. Mz (P’¢ + PM), and the conical fur- 
face belonging M'r =p. M'r(MP'+ Pr), now P ¢ 
+D.y.Aa, Pr=y —D.y' Aa, Mt= Ax 
va + (D. A )s M'r>=Avx /(t 4+ (D ¥')*)s let the 
expreffion V(t +(D. y)’) = 4, then fince y! is what y 
becomes, putting fora,7 4+ Ax / (x + aa ee 
(e+ rd 3 ar pi furface is < p. wba (27 + 
Dy.4x), >p-Aa.) (w+ Ae). y “Dy. Ax); 
let “he function of x = preening the furface be V, then the 
portion belonging to M M’ = 2V. 
Ax)? + Div x)? + &e. hence as eat quantity 
being co onfta: antly rele bet 
tioned, the difference between it-and one of in Tee muit be 
lefs than the difference betweenthe limits, er p.J # ((2 y+ 
D.y.42)—D.V) Ae— Dy (Ax)? — &e. < fe 
Az {le(zy + Diy. Ax)—b (e+ Ax). (zy — 
V.Ar+% 
Log. a. zy—D.V).Ax+a.(Ax)}? 
+B. (Az) + &e. < a's (Az) +H. (An) + Bee 
(a, a', 6, b, &c. reprefenting co-efficients affected with 
A 
powers o of A *) for fince) (w@ + Av) =bax +Die. 
#+D hax a)? ee. and 2y=2y4+2D.y. 
rae “Ke, the firf term in the Pec feries, for : 
(# + Aw) (2y7' — Dy. Ax Jb ae 
rft term 
the expanded feries for} » (2 y ~~ D ye Ax); and ee it 
follows, that the firft Haman 3 Ax. fy w.(2y+D.y. 
Ax) —-L( 2A +2) (2 y' —D.y'. 4 a)t contains (A 
x)* 3 confequently, — the difference between the es 
-AQx 
and D.V, or (g.bar.2y — x +a 
+ &c. mutt be lefs ie the differenee bowen the- Linde. of 
t 
forma! (A-w)* +-8' (Ax) +. = pk {mall A swiss 
this cannot happen.except p.1.2. — D,. Ve =. 0, or 
dV dV: 
P wba. 29 a? or p. 04.0) a9 =H 
orp. (da 4dy). dV; Vv (dx? + 
dy), dz,dV2py d a the diff eel ee from 
ne the. furface is tobe’ found, 
aF oar i ive $dy)= 
dy 
If yoaw dea 
