C szsz, sE—y, CE; 
FLIJX 
and as the tria'ngles CE/, CSY, 
r::j: 
fimilar, 
ry 
are 
&>— hence, SCw; 
A. Alfo, by fi¬ 
at 
niilar feftors S oz, S Cn 
yw 
a 
•y 2w — A. 
a:y \ \w\ x—'— ’> there¬ 
fore SC«: 
2 a 
Thefe different expreftions of the fluxion of the area are 
to be ufed as may be convenient. 
Ex. i. Let SWC be the logarithmic fpiral ; to find its 
area. Here r : t in a conflant ratio, as m : n \ hence, h — 
d— - r -f ...r-r, ti nflnfr to a — i f 1 , hut when y—o, 
Arzo, .•.C=o; confequently A=. 
whofe fluent is A=—-j-C 
2 1 z n 4 u 
my 2 
4 n 
Ex. 2. Let SWC be the fpiral of Archimedes; to find 
its area. Here y : to :: m : n, or in a conftant ratio; .-. 
- ny 
-, confequently A= 
m 2 a 
A; 
A: 
ny 3 
6ma 
ny 3 
6ma 
2ma 
Cj but when y—o, A—o, 
whofe fluent is 
. •. C=o ; hence, 
of the feftor whofe arc is 
Ex. 3. Let the fpiral be a circle; to find its area. Here 
yx. yx 
: —IS A —— 
2 2 
hence, if x =. the cir¬ 
cumference c, the area of the circle — —. . 
2 
Ex. 4. Let AC be the involute of the circle AD, de- 
fcribed by the extremity C of a firing unwinding itfelf 
from tlie circle ; to find its area. 
It is manifeft that DC muft be 
perpendicular to the curve, or 
to its tangent CY, and as SD 
is alfo _j_ to CD, and SY to 
CY, SDCY is a parallelo¬ 
gram, and SD=CY=:/; hence, 
ryy 
21 
SY=r =s /y 2 —t 2 ■ .-. A—- 
y*^i 2 ]jXyy 
2 £ ’ 
by Art. 39. is An 
whofe fluent, 
__3 
y*—t 2 ] 
+ C; 
but when y (SC) becomes t (SA), then A, or SAC, 
6 1 
is 2=0, and_y 2 - 
_DC3 
LbD’ 
hence, C=o; .■. SAC— 
7 " 
6t 
To find the CONTENTS of SOLIDS. 
Prop. XXL —To find the content of a folid generated by 
the rotation of a curve about its axis , or by the motion of a 
•plane parallel to itfelf. 
52. Let the folid ACD be conceived to be generated 
by the uniform motion of the circle CD, beginning at A 
and increafing in magnitude, having its plane always per. 
pendicnlar to AB, and its center in that line. Circum- 
icribe this fohd by the cylinder MLCD, conceived alfo 
to be generated at the fame time by the fame uniform 
motion of a circle. Then AL being conftant whilft BC 
2 - 
I O N- S. 4&1 
varies, let the circle CD move en to mp , and the folid 
Cm.pD generated, will be the increment of ACD ; fup- 
pofe alfo the circle CD to move on to cd in the fame time 
without increafing, and it 
will generate CDdc the 
cotemporary increment of 
the cylinder; produce CD 
to n and q, meeting inn 
and pq drawn parallel to 
BA. Then the ratio of 
the increment CDdc of 
the cylinder to the co¬ 
temporary increment CD 
pm cf the folid ACD, 
is always nearer to a ra¬ 
tio of equality than the 
cylinder CD dc : the cy¬ 
linder mnqp , or nearer 
than BC 2 : bm 2 . Now, 
let the circle mp move up 
to and coincide with CD, 
in order to obtain th e limit¬ 
ing ratio of the increments, and we get the limiting ratio 
of BC 2 : bm 2 , a ratio of equality ; hence, a fortiori, the 
limiting .ratio of the increment CD dc of the cylinder, to 
the cotemporary increment CD pm of the folid ACD, is 
a-ratio of equality; therefore by Prop. 2. Cor. 1. the 
fluxion of the cylinder MLCD is equal to the fluxion 
of the folid ACD ; but CDi/c being the increment 
of the cylinder uniformly generated, will- reprefent 
its fluxion, by Prop. 1. hence, the fluxion of the folid 
ACD will be reprefented by CD dc, the cotemporary 
fluxion of AB being BA. Put therefore x—AB, y— BC, 
Sr=: the folid ACD, p— 3,14159, &c. then (Art. 
49. Ex. 2. Cor.) py 2 — the area of the circle CBD ; 
hence, the cylinder QDdc—py 2 x— S ; therefore S=: the 
fluent of py 2 x, corrected if neceftary. The fame reafoning 
will manifeftly hold, if the generating plane beany other 
figure, and continue always parallel to itfelf. The 
fl uxion therefore of a folid thus generated, will be always 
ex prefled by the area of the generating plane and its 
velocity conjointly. 
Ex. 1. Let ACD be a folid generated by the revolu¬ 
tion of any parabola about its axis. Here ax—y "; hence, 
... 
whofe fluent is S : 
wpy ,,J c 2 _ n 
■ - +c=- x Py 
k + n 4- 2 
y n ^ n 
X --K— -T~ 
a » + 2 
X py 2 x + C 
but when x—o, S—o, 
•.C—o; hence,S=—;—X/tr 2 *. 
«—{- 2 
If nzzz2, the folid becomes the common paraboloid, 
and its content — \py 2 x—± cylinder LCDM. 
If n— 1, the curve becomes a ftraight line, and the folid 
a cone, and its content = § py 2 —^ cylinder LCDM. 
Ex. 2. Let APEQ^be afolid generated by the revolution 
of an ellipfe AEPQabout its axis AE. Put AB=x, BC=y, 
AO—a, FO—b ; then, 
by the property of the 
eliipfe, a 2 : A 2 :: 2ax —.v 2 
b 2 -, 
: y 2 =-—X 2 ax —x 2 ;hence 
py 2 x 
pb 2 
x H 
2axx — x 2 x, whofe fluent 
pb 2 - 
is S ———■ X ax 2 —4-x 3 4- 
c 
r 
j3 • 
r 
0 / 
C ; but when xz=e o, S; 
.•.C=o; hence, -v 
a 2 
ax 2 —which is the folid content of ACD ; and to get 
the whole folid, we muft make AB equal to AE, or make 
XZX.2R1 
