490 FLUX 
Cafe 3. Tn refnedt to the at'ea BCM, as this area de- 
creafes by the fame quantity tliat ABCP increafes, it 
will Iiave the fame fluxion, only with a contrary fign, 
by Art. 16. hence, the fluent will be the fame with the 
JC ^ ‘ 1 '" n 
fign changed, that is BCM=F-EC. If n be greater 
n —1 
than unity, BCM - - - -j-C; and when x is infi- 
n— 1 ,x H 1 
nite, BCM—o; hence, c— - ---j- C, and there- 
n —1 ,x n — 1 
fore C—.—■ . ; ■ ,' -=0, .vbeing infinite ; confequently 
n — i.x :: — 1 
BCM —--- 
n —1. x" — 1 
C;tfe 4,v If n be lefs than unity, and x become infinite, 
JQ 1 - 1 
C— --, an infinite quantity j hence, the area BCM= 
t —u . 4 
-!-C is infinite. 
n —1 
Cafe 5. If n— 1, this fluent fails (Article 38), and the 
hyperbola becomes the common hyperbola. Let AB— 
BC=zi, BR:=x, RS—jv, then AR=t -\-x, and y ———> 
* 
X 
therefore the fluxion of the area BCSRz=- whofe 
1 -\-x 
fluent, by Art. 45. is the h. \r i-\-x, which wants no 
ccrredtion, becaufe when x—o, the area BCRS=o, and 
the fluent becomes the h. 1 . 1 which — o. Hence it 
•appears, that any area BCSR is the h. 1 . of the abfeifla 
AR. The modulus is here unity. 
Ex. 4. Let MCD be the logarithmic curve; to find 
its area. The property of the logarithmic curve is this, 
that if the abfeifla AB increafe in arithmetical pregreflion, 
the ordinate BD will increafe in geometrical progreflion ; 
if a—AB, t=BD, a—AC, then, by Art. 44, M= —’ 
y 
■which, (by Article 23.) is the fubtangent AT; lienee, 
A — vr —M y , whofe fluent is ArrMy+C ; but when 
y—a, A—o, .o=M«-fC, and C——M a; confequently 
AB-DCrxMy—M«=ATx BD—AC. Hence, the whole 
area DMB=ATxBD, becaufe at an infinite diftance 
AC=o. 
Ex. 5. To find the area of the catenary curve ACB. 
Put CE=r, *EF =y, CF—z ; then z 2 —2ax-\~x 2 , and 
zz=ex-\-xxj hence, z 2 z 2 =za- > r x X-* 2 ; but z 2 ~2ax~\-x 2 
— ei + x — a 2 > an g y.i 
—z 2 —y 2 (Prop. 23); 
hence a-\-xX, z 2 — a 2 z 2 
— a-\-x X z 2 — -j 2 , or 
a 2 z 2 —a-\-x xj 2 ,and 
az—a-\-x X.y—ay-\-xy; 
hence, xy — az — ay-, 
but flux. xy~xy-\-yx ; 
therefore xy—flux. xy 
—yx; hence, flux, xy 
r—yxzzai — ay, and A=yx~flux. xy — az-fay ; therefore 
I O N Sr 
A— xy — az-\-ay-\-C ; but when x—o, thenjy:=:o, z~ o, and 
A=o ; therefo re C—o ; lienee, A =zxy—az+ay=a-\-x X 
y—a f 2ax-\-x 2 , the area CE 1 1 '. 
Ex. 6. To find the area of the cycloid ABC. Let 
BD be the axis, on which deferibe the circle B/iDai, 
Bn. Now by the property of the cycloid, the triangles 
B rn,yzv, are fimilar ; hence, Br, or ty, : rn :: zv, or rq, : 
zy, .-. rhy.rq-tyy.zy, or cn nrqm— cn styz, that is, by 
Art. 49. the fluxion of the circular area Bur — the fluxion 
of the area Bty; and as thefe areas begin together at B, 
and their cotemporary fluxions are always equal, the 
quantities generated are equal; hence, the area Bty—the 
circular area B;ir; bring therefore yr down to AD, and 
we have the whole area BFA =: the femicircle B«D ; 
hence, BFA-j-BEC = the whole circle BnOzo. Now the 
parallelogram Al'EC— ACxDD= (from the nature of 
the cycloid) circum. B«Dro 3 x BD— (by Art. 51. Ex. 3.) 
four times the area of the whole circle ; hence, ABC=: 
three times the whole circle. 
To find the AREAS of SPIRALS. 
Prop. XX .—To find the area SWC of a flpiral. 
50. Let SWCK be a fpiral, generated by the uniform 
angular motion of SC about S ; SC any ordinate ; with 
the center S deferibe the 
circular arc XCZ ; draw 
any other ordinate Sv, and 
with the center S deferibe 
the circular arc wwmeeting X 
SC produced in w. Now 
conceive the fedtor SXC to 
have been generated by the 
uniform angular motion of 
its radius about S, at the 
fame time that the area 
SWC of the fpiral was ge¬ 
nerated by the fame uni¬ 
form angular motion of SC 
about S. Then SX being 
conftant whilft SC varies, the increment of the fedtor 
SXC is the fector SCn, and the cotemporary increment 
of the area-SWC of the fpiral is SCr/; hence, the ratio 
of the increment SC?z of the fedlor SXC to the cotempo¬ 
rary increment SCw of the area SWC, is always nearer 
to a ratio of equality, than SC n : Swv, or nearer than 
SC 2 : Sw 2 ; now let S<y move up to and coincide with SC, 
in order to obtain the limiting ratio of the increments, 
and we get the limiting ratio of SC 2 : Sv 2 , a ratio of equa¬ 
lity ; hence, a fortiori, the limiting ratio of the increment 
SC» to the increment SCw, is a ratio of equality ; there¬ 
fore by Prop 2. Cor. 1. the fluxion of the area of the 
fedfor SXC is equal to the fluxion of the area SWC of 
the fpiral; but S Cn being the increment of the fedtor 
SXC uniformly generated, will reprefent its fluxion by 
Prop. 1. hence, the fluxion of the area SWC of the 
fpiral will be reprefented by CS«. 
51. Put SC—y, the length of the curve SWC=z, 
XCz=x, Cn=x ) A 2= the area SWC; then the fector 
SCs=— —A, whofe fluent is the area SWC. Let sCY 
2 
be a tangent at C, and SY perpendicular to CY; draw 
CE JL SC, and sE parallel to SC; and with the center S 
and any radius SA, deferibe a circular arc AL. Put 
SA =«, A o-zzw, ozzzzo, CY=9 SY=r. Then by Art. 31. 
C szzZj 
