: 3 FLUXIONS. 
Fig. 36. 
and (= ET f(Gds—2¢ eds sds), 
453 
Inverse =—- Exapye 1; Let the ing curve be a line, which, passing through A, is carried round in the _ Inverse 
= Se IeasrEne ths cr aaaeutais Greanference of he PRY p’. Method. 
Let.AQB, a perpendicular to the base of the solid, 
meet PE p, a section parallel to the base inQ. Put the 
area of the base = 6, the perpendicular A B = p, and 
wehitsves, be paler (avai pees tae heap eons 
whatever, by a plane | to its ing always 
similar to the bust, we have prati:b: X, hence 
523 
3p?- 
Here no correction is wanted, because when z= 0, then 
s=0; and putting X instead of =% 
} Xz, but X 2 is the content of a cylinder whose base 
is X, and altitide x; therefore every solid of this kind 
is one-third of a cylinder of the same base and altitude. 
This rule applies to cones and pyramids, whose bases 
are any whatever. - 
Ex, 2. Let the solid’ be what is called a Groin 
(Fig. 38.), which is 
cefg moving 
Xde= fedex 
» we have s= 
ee by 
to itself, the section DAH, 
and taking the fluents of the several terms, the je of the opposite sides, being a semi- 
we AMO et on ied AB ical to the plane of the 
$= — (9tr—igtz3 2) Le , put AB=p, AQ=z, PQ=7; then f=2 pxr— 
~~? aSAns FEE Fall: by the nature of the circle, but 45? ia the area‘? the 
If the fluent commence when r=0, thene=0: And section ce fg=—X, hence " ‘ 
x 
z 
duced by the revolution of an ellipse about 
DE=3, also AQ=s, QP=y. 
ellipse, y*= = (az—=2"), hence 
making 4 =<. We gato P! a= ff the content of 
1st Ett site i APD, (ig 9 pro. 
fixed 
a, the revolving axis 
rom the nature of the 
safXde=f(Sprdze—s2°dz)=4p2——_. 
no correction is wanted, because if z—=0, then 
és 
= When «=p, then s=£p3, the content of the 
hole’ solid ‘P> P* 
Of the Surfaces of Solids. 
a variable square Fig, 38. 
_ eh, «bt 
=f G@r—s)de= = (fas*—425) 4. 163. In Fig. 20, let AB be the axis of a solid of res of the sur- 
volution, and AP the oe ree Bae put AQ=z, faces of so- 
PQ =y, the arc AP = =, and the'surface generated by lids. 
the curve AP=v. The general formula is in this case Fis- 20. 
3 supposing a = 6, have § ra? 
content of a sphere, > debater isa A et gee ede 9 
is the area of a section of the spheroid v=2n fyo/(der+dy)=2" fydz 
the centre, it ‘that: the ~whole solid is 3 of We shall now apply this to some exam: 
Exampce 1. the solid be caohie at which the 
Ene ne nae eae renee te Gael be Pig, 34, 
r . surface of segment, tpl 7 AW gre oT ca 
etx + ama pendicular i of the sphere 
162. If the solid APE (Fig. 19.) is not formed bythe °° SMaWPO sili uadinenie 20 
teen & Waid; yot if theca =a, AQ the height of the =2; the radius of 
"be eferred to an axis AB, so that PE p, airy section of  '° P&=y; by the = oe Netiwue 
the solid by a to axis, issome %42—x2*, hence ydy=(a—x)da, and dy= ’ 
known function of AQ = z, the segment of the axis be- 
tween the plane and a given A, its content 
le found fen the wiry taasd fornonle Bor ©. aiay-bo 
as m art. 79, that a ratio of equality is _ hence, du=2ryi/(d2*+dy)=2 rade, 
and vw=2 raz. 
2, 
8, 
F 
E 
: 
. equal to a rectangle contained by the height of the 
a ever and. a given point out of it ment, and a straight line equal tthe circumference of 
" Sito warn a straight that the whole surface of the sphere is four times the 
