Chap. XVI.

HYDROSTATICS.

Ill]

Let A B C D be a right cylinder cut into two equal partsby a plane E F G H passing through the axis I K, then byanother R O B M which forms an ellipsis, lastly by twoother planes parallel to the base, forming two circles, ofwhich the first, O L M N, passes through the small axis O Mof the ellipsis, and the second, OPVII, through the ex-tremity R of the great axis.

Let us consider that all these sections produce severalsolids:—firstly, the frustum ROM NR, formed by thesemicircles OMN, the semi-ellipsis OM R, and a portionof the surface of the cylinder M V N RT O.

Secondly ; another frustum, OLBM O, equal and similarto the preceding, since it is also formed by the semicircleO L M , the semi-ellipsis 013 M, and a portion of the surfaceof the cylinder OLMB.

Thirdly; the solid RQOMVR, formed by the semi-circle RQV, the rectangle QOMV, the semi-ellipsisO M R, and the two portions VMVRV, Q O T R A of thesurface of the cylinder : I shall call this solid the comple-ment of the frustum It O M N, because it is the part whichis deficient to make the half cylinder QOMN11V A.

Fourthly ; the solid B M R O Q P B, formed by thecircle PR, the ellipsis O M B R, and the portion of thecylinder comprised between these two planes.

Fifthly ; the solid ABMRDE O 13.

Sixthly ; the two solids O E A B M H E and O E D R M H E.

If we examine each of these solids in particular, we may con-sider the frustum OLBMO as composed of an infinity ofrectangles * D E F C, which would have for their base thedouble ordinate C F of the semicircle OLM , and for heightthe corresponding element GII of the right-angled triangleilliX; thus we shall find the sum of all these rectangles in thesame manner as we find the solidity of the frustum.

We may also imagine the frustum composed of an infinity ofright-angled triangles FHG of an infinitely small thickness,having for base the ordinates F G of the semicircle, and forheight the corresponding element F H of the surface : to havethe sum of these triangles we shall call the radius D L orDO, a; the height, LB, b; D G, x; G F,y ; G,g ; K f,

fix', FK ,dy, and FH,

a

Multiplying the triangle FHG (

will

give

byydx2 a

aab~‘ 2 ~ '

2

aab

2a

Fig. 1692.

Fig. 1693.

by d x, the product

for the differential solid of the frustum; and

as the property of the demicircle gives a — xx==yy, substi-tuting the value of yy in the preceding expression, we shallabdx bxxdx , . nbx bx 3

nave —----, the integral of which is-, or

6 2 6a

Fig. 1694.

aab

When x becomes a we have for the solidity of half the solid, conse-quently ——— for the entire solidity, or when a =b, which shows that in this case the3 3

frustum is equal to the third of the cube of the radius.

From what has gone before, we may regard the infinitely small superficies FH fhas the differential rectangle of the surface of the frustum, of whose base we shall havethe expression by drawing the radius D F, and considering that the similar triangles

F1)G and F/K give F G (y) F D (a) : •. f K(dx') f¥ of which the fourth term

being multiplied by F FI —) gives ^¥ - a , or simply bdx, whose integral is bx or ba

when x = a for half the surface of the frustum, consequently 2a6 for the entire surface,which is found equal to the rectangle comprised by the diameter M O and the height B Lof the frustum.

4 b 4