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Example IX. In the logarithmic spiral

cos Mmr : mr ::1: Mm that is c:y::1:

y

2 Therefore Adm=y=MT:

=YM

с

B

с

ON THE CUBATURE OF SOLIDS.

114. In order to find the contents of any proposed solid, we must conceive it to be decomposed into an infinite number of thin segments or slices parallel to one another. Calling t the surface of one of these segments, its thickness or an indefinitely small portion of a line perpendicular to that segment; S (10)+C will be the content of the proposed solid; there will then only remain for us to find the value of t in terms of x.

115. For example, let B be the base of the solid ; H its altitude, or the distance of the base from its vertex. If we suppose that the surfaces of these segments are in proportion to any power m of their

B.x** distance from the vertex, we shall have H" : B:: x": Conse

H" quently the solidity of any portion of the solid will be C+ Bx**

Sm+1

m+1 Br

B.x ; or simply

if the portion begins from (m+1) A"

(m+1)
HTM'

BH the vertex. Consequently the entire solid = (because x coin

m+1' cides with H). Therefore in pyramids this solidity = BH, since m=2.

116. If any curve AM revolves about its axis AP, it will generate M

ai a solid of revolution of which each section perpendicular to the axis will be a circle, having for its expression myy, calling PM = y, and 7=3.14159, &c. Consequent

BE ly the content of any solid of revolution =C+jayya.

C+

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111,

12'

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Example I. In the sphere yy=2ar-rx. Therefore the solidity of any spherical segment = xx (a-4x), and the sphere = 2a?r= two-thirds of the circumscribing cylinder.

_bb

Ex. II. In the ellipse yy=2 (2ax-x.x). Consequently the solid

aa

m

generated by its revolution about its transverse axis is to the circumscribing sphere as bb : aa; or it is = to two-thirds the circumscribing cylinder.

117. The solid which we have just considered is called an oblong spheroid ; and that which is formed by the revolution of an ellipse about its less or conjugate axis, is termed an oblate spheroid. It is easy to shew that this latter solid is also equal to two-thirds of its circumscribing cylinder. Consequently the oblong is to the oblate spheroid :: abb: aab::6: a.

Ex. II. If a parabola of any degree whose equation is y=x" a"-* revolves about its axis, it will generate a solid, having for its expression Sayo i= Toxy? ; or which will be to the circum.

m + 2n scribing cylinder :: m: m +2n. Thus the common paraboloid, in which m=2, n=1, is the half of its circumscribed cylinder.

Example IV. Similarly, if the hyperbola whose equation is y" ..." mtn

revolves about its asymptote CP; by taking CD=AD=a, the solid described by the trapezium ADMP, will be expressed by

-# (a--xy?), and consequent2n-m

P ly the solid described by the indefinitely long space OADX, is to the cylinder described by ACDE :: m: 2n—m; and in the common hyperbola it is equal to such cylinder.

=a

M

m

ON THE CURVILINEAR SURFACES OF

SOLIDS OF REVOLUTION.

N

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118. The Auxion of the surface described by the curve AM= is equal to the surface of the little truncated

М.

M cone described by the element of the curve Mm. Consequently the curve surface = SMm circ PM=2 asyn (i?+y)+C=2 fix +C, by making

BE the normal MN=n.

Ex. I. In the sphere n =a. Therefore the surface of any spherical segment = 2 anx, and that of the sphere =4 a' =4 great cirdes. Ex. II. In the paraboloid, because n=v(yy + 1 pp), the surface

27 pa

Зр

3

.

Let y=0,

and we shall have CE

and the surface of the solid

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Зр

will be: S 299 (99+£pp)*+c= (y*+679+c.

Šp ={(up+499)} y = = v(aa-xx). Therefore

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-p8

Example III. In the ellipse 6

—.

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a

B

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a

f-iw {

{ aam(aa-bby me, and

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aa

E as a is the axis of revolution, it will denote the semi-transverse axis in the oblong spheroid, and the semi-conjugate axis in the oblate spheroid.

In the first case let aa-bb=mm, and we shall have

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-*)

. Consequently if with a radius CD= =

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a :

we describe a

m

m

circular arc DBN, we shall obtain for the expression of the surface

2b7m described by the revolution of AM about AP,

X ABNP.

aa

In the second case, making bb--aa=mm, we shall have

at bamx

XX

+

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aa

m

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aabr

= the surface described by the rem volution of AM about CE. Here we must observe, that CE-a, CA=b, CQ=x, QM=y.

6

Ex. IV. In the hyperbola y= V(xx-aa)

2

an

m

Therefore if this curve revolves about the axis AP, the surface described by the arc AM, (by making aa+bb=mm, and determining the

2brm constant) will be

"g*)(**

g
bama

arby
W (or-
xx -bbm

log
mx + (mox?-a^)

a (m+b) And if it revolves about its second axis CQ, then y=MQ=7 (66+xx). Therefore the surface described by the arc AM= ambb

mox?
(xx+2) + log { het + v(1 +ment)}

an

m

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ON THE INVERSE METHOD OF TANGENTS, AND ON FLUXIONAL

EQUATIONS.

119. The inverse method of tangents is that which teaches us to discover the equation of a curve from some known property of its tangents.

120. For example, let us investigate the curve in which the subnormal is constant, or =a. Since we already know, that the gene

ral expression of this line is yy, we shall have 99=a, yj=ai; and

integrating, in order to express that the given property belongs to every point of the curve, we have yy=2a (x +c) an equation to the parabola which solves the proposed problem.

121. The inverse method of tangents is always reducible to the solution of a fuxional equation. As we have not yet treated of these sorts of equations, before we proceed farther, it will be necessary to explain something of the theory of them,

Equations, in which only first fluxions enter, are called fluxional equations of the first order. Fluxional equations of the second order, are those which contain second fluxions, excluding fluxions of any higher order than the second; and the same with other orders of equations.

122. In general let P and Q be any two functions of the variables x and y; then P.: +Qy=o, will represent generally every fluxional equation of the first order, containing two variables x and y, and it is evident that it will be integrable. Io when both P and Q are functions of x or of y alone; and also

y Р II° when we have

123. But when these conditions do not take place, we must endeavour to separate the variables ; that is, to divide the equation into two members, each of which shall contain only a single variable and its fluxion. No general method of performing this separation can be assigned; we shall however give a few cases in which it can be done. 124. If P=XY, and Q=X'Y', X and X being functions of 2,

Xi Y and Y' functions of y, we shall have

Y which the variables are separated, and which therefore is reduced to the integration of fluxions containing only a single variable quantity.

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Yy, an equation in

X?

Examples 1. Required the relation of x and y in the fluxional equation "x+y" y=0. 2. Solve the fluxional equation yr-xy=0. 3. Integrate the equation y+by* x=ax. 4. Solve the fluxional equation myx +nxy=0.

a (mæ+yy) – yrxy +36 yosj=o.

yx 5. Integrate the equation

+ ✓(r+y x + y*

and 125. If P and Q are homogeneous functions of x y,

that is, if every term contains the same number of dimensions of x and y;

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