Fig. 39. 
of one of its great circles. Both of these conclu- 
sions were found by Archimedes. . 
congas reabry tees gi me Noe a A 
19.), then putting AQ=-+, PQ=y, the parameter of 
- eer saiennenee (art, 157. Ex, 2.) that 
¥de+dy= ~ dyvi(y? +a"), hence 
de=tay/ (de +dy)= SF ydyVly +0") 
and taking the fluent (art. 122.), 
= Sa 
If we we suppose the fluent to begin when x=0, then, 
because y is also =0, we have O=}ra? +o; hence 
c= —} 7a", and the adjusted fluent is 
sale {(fser—o} ’ 
164. We have observed (art. 107--110), that the inde- 
finitely small increments quantities may often, on the 
grounds of convenience, be taken as their fluxions ; and, 
indeed, this is in effect the same thing as to seek the 
limit of the ratio of the Ente Sortiateeh. and then to 
consider it as the ratio of the fluxions. We shall now 
give two examples of this application of the infini- 
tesimals, 
Exampce 1. Let it be required to find the surface of 
ue cone, (Fig. 39.) 
be the centre of the base, V the vertex, VA the 
from the vertex, pent re pend in =H 
pert the civeumfarers 
take B any in the eineninfarenen aree 
draw the nt BD, CA_ produced in Ee 
draw VD icular to BE, and join CB, BA, DA, 
VE. The triangles VDE, VAE, are right-angled at 
DandA; 
AE? — AD? = VE* — VD? = 
“‘DE2; therefore "AD is 
porpenslictlar to EB, and uently lel to CB. 
us su) the radius of base to, be 1; put a 
fe fe istance of the from the cen- 
p for VA the altitude; and ¢ for the variable arc 
HB. P The trian les ECB, "BAD, are similar; hence 
EC: CB:: EA: AD; that is, sec. 9: 1: 7st. O— a 
sec. 0—a 
:AD = sec. @ = 1—a.cos. @, therefore,, 
DV= were acos.e) hs 
Take a point 4 in y near to B in the circum- 
ference of the base of. the cone; and join C 4, Vb; the 
small arc B 4 may be considered as coinci with its sma 
tengint. We are now to consider the indefinitely little 
arc Bd as the fluxion of BH=¢9,-and the VBé 
as the fluxion of the conical surface which the line VB 
poe over, while it moved from. the, position) H, 
along the are HB. We have therefore B J=d 9 ; and, 
as the area of the triang! Pee Nahe g ks 6x VD, if we put 
¢ tr td eee a ie tons oe, 
dsx}doy fr+a 2u'@ con Oe a 
This is the expression for the fluxion of any conical, 
surface whatever, having a circle for its base. The des, 
termination of the fluent has long exercised the i 
of mathematicians ; and we .observe, that 
has at last; succeeded in expressing the whole 
app eeny (that is, the fluent between. the li- 
mits of @ = 0, and = 2 #) by elliptic arcs, ( Ezercises 
FLUXIONS. 
ore VE? — EA!—VDt— DA?;hence 4 
2 0 cm'gul 9 il be te phere? surface APQ: 
de Cal. I 178). Bat the indefinite Aegot, ioe the 
n for the nace 
whatever, has not-been 
the conic. sections, pe for pa 
and p.° 
t f rf my 
cd 
If make cos. shen a sig rig tp eat Dash 
we ox oe Va—s) 1s)! 
The fluent be. fi 
Ph ge fy i 
on Bee A BT On UM A ate pres 
ds=4do FSi ty ve 1 ae 
the surface is Mire frodack of 3. acre fe 
sant side ofthe cone 
x. 2, Su circle to be di a Sm 
radius of the base of 1 eseibed aru 
an upright cylinder to be raised, 
here the hemisphere ; it eee 
the oval Hole made in the a the su 
which is bounded’ 
one of whichis DPAQD. 
Take P ae ate 
plane to pass tle re pose 
throu, 
P, m Spl the 
eee DPF, its its Sask me ocr: and fe 
of the cylinder in the straight line PE. Let CF meet 
rT es the base of the cylinder i in E, j join 
an 
_, The right angled triangles CEP, CEA have the 
sides, CP, CAy. ite .to the _ angle in: each, 
equal, and the side GE common to both; therefore the 
triangles are, equal, and the angle ECP:is equaltothe 
e ECA,. are FP is equal to the are FA: 
ee D. — pa ater eramnE nas UAE A 
and suppose another: ‘circle pq to 
ap fen indefitly near to the:former.»: We may 
consider the surface eontained»betweenA-PD and 
wh <Deatl.the capmen aeadehiod whieh 
as gq f 7 ‘ 
meters Put a for the radius of the sphere, and 
be ae > ACF or FCP ; then: the: AF, FP, 
and. AQ _ will each be, tae go to: ag; and Qq will res 
pendrin nt ears the are AQ;; and because 
the radius. of the small circle PQ iscw cos. 9,°we have 
a: aos, 9:29 (mare AF): PQ, hence PQ=a¢ cos. 9; 
and the area «PQ q p,. (= _—a xr ) Oss 
AF OP CO%.9 dQ; shenefucethasiads all the areas; 
(arre ts 
(art. 153,) “This fluent is a? (, 
but when ¢ = 0, then, the ; 
a = — a’, and,.the- <y" n 1 to 
sin, @—1). —_ 
gunn (4) giver {at a aune 2 a ‘ 
wi '} 
area of ehaeoal Nelolle “vault. - So 
165. In the year 1692, Viviani, ane of Galileo's dis« 
