493 
x-=2a\ hence, the whole folid 
FLUXIONS. 
ph* 
X4 a3 —1“ 3 
~r 4 pb 2a . 
If the eilipfe revolve about PQJnfteadof AE, then, as the 
fame property of the curve holds for each axis, the folid 
will be —-. hence, the folid generated about AE .-folid 
about PQ-.. ::^:<z:: p CU AE. Iffc=a, the 
‘ 3 3 
cllipfe APE(Vbecomes a circle, and the folid a fphere, and 
A. i) b 3 
the content becomes — -i — —y,i%Zy)b 3 . Now the con- 
3 
tent of a cylinder circutnfcribing the fphere — the area 
of its end multiplied by its length — (as the radius of the 
end —b, and length —2b) pb 2 x xb—xpb 3 j hence, the 
fphere : cylinder :: : x :: x : 3. 
Ex. 3. To find the content of the folid generated by 
the revolution of the cifloxd of Diodes about its axis. 
x 3 * 
The equation of this curve isy 2 ^-—-5 hence, S —py 2 x 
_ x __ _ /p, fl; v ifl on ) — pa 2 x—paxx — pa 2 x- J- 
CI—-X 
; now the fluent of all the terms, except the laft, 
a—x 
is found by Art. 37. and the fluent of the laft, by Art. 45. 
hence, the fluent is S=— * 3 -px 3 —%pax‘ — pa*x+pa 3 X — h. 1 . 
a —x + C ; now, when x—o, S—o, x.pa 3 X —h.l.^-f- 
C=o, and C —pa 3 '/, h. 1 . a : hence, S = — ! s px 3 — ± pax 2 
+pa 2 x+pa 3 X"— b. 1. a — x+pu 3 X h. 1. a — — \px 3 — 
| pax 2 — pa 2 x-{-pa 3 X h. 1 . ; becaufe h. 1 . a — h. \.a —at 
s=h.I. 
-, by the nature of logarithms. 
Ex. 4. To find the content of the folid generated by 
te logarithmic curve ABDC revolving about AB. 
Here yx = My, 
by Art.49. Ex 4. 
.S —py 2 x — 
Mpyy , whofe 
fluent is S — 
Sld+Cibn, 
2 
when yzza, S—o 
Mpa 2 
.-. o=—— + 
2 
C,and C— — — j hence, S= —X 7 2 — If AC = 
M py 2 
— the whole folid Correfponding 
e—o, then S: 
to the abfcifla BM. 
Ex. 5. Let the catenary curve revolve about its axis, 
to find the content of the lolid generated. By Prop. 108, 
of Vince’s Fluxions, z 2 — xax-\-x 2 , and therefore zz—ax 
-j-xx; and by the fame Prop, zy—ax. Now Szzj0y 2 x; 
aflume therefore S —py 2 x-\-w, and we have S— py 2 x -j- 
xpxyj w, and as S — py 2 x, we have zu— — xpxyy— 
/ . tix\ 
\ y ~~z ) 
xpay X — = (as xx — 
• ax) — xpay 
Ex. 6. Let the conchoid DM of Nicomedes revolve 
about the axis DA ; to find the content of the folid ge¬ 
nerated by DMF. By Wood’s Algebra, Art. 497, if 
CA — a, AD= EM~£, AP = at, PM —y, then x 2 — 
l ±L ~ X , t : - y2 ; alfo, D 
y 2 F 
px 2 =;the area of the 
circle generated by 
FM, and as FD=6 
— y, FD^r—hence, A 
pyx 
S~ — px 2 j — 
a -\-y ——— 
~jy-Xb-—y 2 
■ px 
a+yx j —pa 2 b 2 y~xj—p[ > 2 
pa 2 b 2 
xpa!> 2 y , * 
-—, therefore S—~x 
y 3 
a+y +- 
7 
pb 2 y—■ xpab 2 x h.l.jy-f-C; now when 
Ay — -- 3 
y—b, S:=o, and the equation becomes cz=.-xa-yb + 
pa 2 b — pb 3 — xpab 2 y.\y. 1 . b-\-C, therefore C; 
p -s 
■ ~ X ci-fi b 
3 
p -- 3 p 
— pa 2 b-\-pb 3 -\-2pab 2 x h. 1 .b; hence, 
3 3 
-3 ta 2 b 2 b 
a-\-b -j- —- P a2 b — pb 2 y+pb 3 -\-xpab 2 x h. 1 , - the folid 
generated by DMF. 
The folid generated by the whole curve is infinite, 
as appears by making y— o. 
Ex. 7. Let LAO be a folid called a groin, generated by 
a variable fquare vwxz moving parallel to itfelf; and let 
the feftion FAG through the middle of the oppofite fides 
be a femicircle. Put azxAE, *=AB, y=BC ; then, by 
the property of 
the circle, y — 
f 2ax — x 2 , there¬ 
fore the fide of 
the fquare vwxz 
— 2 f ax — x 2 ; 
hence, the area 
vwxz — 4 X _ _ 
xax — x 2 , which, L O 
being the generating plane, it anfwers to py 2 in the other 
cafes, and therefore Sz=4X xaxx — x 2 x, whofe fluent is 
S —4 ax 2 —^x 3 + C; but when x — o, S = o, .’.C^o; 
hence, S—4«x 2 —-|x 3 , the folid A vwxz; and if we make 
8a 3 
x—a, S =—•, the whole folid ALN. If the fe£tion FAG 
3 
be any other figure ; or if the two feftions through the 
two oppofite fides be of different figures, the content 
may be found in like manner. 
To FIND THE LENGTHS of CURVES. 
Prop. XXII. —To find the length of a curve line AC, 
'whofe ordinate B C is perpendicular to the abfcifia A B. 
53. Put A B—x, B C—y, AC—z 1 then if Cs be a tan¬ 
gent to the curve, CE j_ 
BC, and /E _]_ CE, we have, 
by Art. 27. CE=x, sE—y, 
Cszzi ,; and by Euclid, B. i. 
X«- = —xpay X z—y-— xpayj—xpayz ; aflume w— P- 47 - z ~ 
+i 2 
pay 2 — xpayz-\-v, then w—2payy — xpayz — xpazy-\-v; and 
as w—xpayy — xpayz, we have v—xpazj—xpa 2 x, therefore 
vxzxpcAx-, hence, S—py 2 x = pay 2 — xpayz-\-xpa 2 xfC ; but 
when x—.0, then_y—o, z—o, and Sc=o, therefore C=o; 
««jnfequently S —py 2 xfpay - —xpayzfxpa 2 x. 
■\/x 2 -\-y 2 , and z— the fluent 
of V* 2J rj 2 , corrected if ne- 
ceflary. A 
Ex. x. Let AC be a femi-cubical parabola, whofe 
equation is ax 2 —y 3 5 to find its length. Here ( ... 
a 2 
