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ON THE CURVILINEAR SURFACES OF

SOLIDS OF REVOLUTION.

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

M cone described by the element of the curve Mm. Consequently the curve

N surface = SMm circ PM=2 #syv (?+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'F=4 great cirdes. Ex. II. In the paraboloid, because n=v(yy + 1 pp), the surface

27 will be

4* *py=

3p

47 and we shall have CE

the surface of the solid 3p

Šp, and

4yy бр

Example III. In the ellipse y=v (aa-xx). Therefore the surface described by the revolution of the arc AM about its

2 ba axis a will be expressed by

B.

a

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a

S-ów {aaaa-bbits, and

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 aa665mm, and we shall have 2bem

aa

m

*). Consequently if with a radius CD= we describe a circular arc DBN, we shall obtain for the expression of the surface

26mm described by the revolution of AM about AP,

X ABNP.

aa

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

bamx
+x)= s(xx + +

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aa

aa

aaba log {=+v*+=}=

= the surface described by the re

m

volution of AM about CE. Here we must observe, that CE=a, CA=b, CQ=x, QM=y.

Ex. IV. In the hyperbola y= V(xx-aa) Therefore if this curve revolves about the axis AP, the surface described by the arc AM, (by making aa+bb=mm, and determining the 2bam

at constant) will be

A
an
bumi

a'ba
7 (xx -bby-

mx+(mox?-a^)

a (m+b) And if it revolves about its second axis CQ, then y=MQ==; v (6b+xx). Therefore the surface described by the arc AM=

64 ambb ✓ (xx+ 2) + +v(1 + is)} bb

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log

an

m

m

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log 36

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m

ON THE INVERSE METHOD OF TANGENTS, AND ON FLUXIONAL

EQUATIONS.

119. T'he 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 general expression of this line is yy, we shall have 99=a, yy=as; and

X 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 fluxional 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 equalions of the first order. Fluxional equations of the second order, are those which contain second fluxions, excluding Aluxions 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 Pr+Qy=0, will represent generally every fluxional equation of the first order, containing two variables 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 P II° when we have

-Q

y 123. But when these conditions do not take place, we must ene deavour 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=XY, X and X being functions of 1,

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

an equation in

X' 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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Examples. 1. Required the relation of x and y in the fluxional equation x" x+y" j=0.

2. Solve the fluxional equation yx-xy=0.
3. Integrate the equation ý+by crac.
4. Solve the fluxional equation myx +nxy=o.
5. Integrate the equation

a (x2+yy) _yxXY

+36 yoj=0.

V(x* +y? x + y 125. If P and Q are homogeneous functions of x and y, that is, if every term contains the same number of dimensions of x and y;

2+2

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an—mo

an

then making *=2, we readily perceive that

to 음 will be some funcy tion 2 of z. Therefore we shall have x+2y=, or zy

zy+oz+ Zj=0,

3 and separating, we shall find

y For example, (ax+by) : =(mx +ny) į becomes y (az+b) 2

on making+=z; an equation easily intey az+(6m) 2_n

y grated by what has been already explained.

Examples for Practice. 1. Solve the fluxional equation *:* +4y=2yc. 2. Integrate the equation xy-yö = v(x* +yo). 3. Integrate the equation yy +(2+2y)==0.

126. Let there now be proposed the equation (ax+by+c) : + (mx +ny+p) y=0; we must first make ax+by+c=u, and

nu-bz+bp--en mx + ny + p =%, and we shall find x =

and az—-nu +mcap; substituting, we have (nu-mz) is +(az-bu)ż=0,

- bn of which the fluent may be found by the last article.

buix Let again axi + byč + smo yu (fxy + gyr) =0

+

y (fiy 1 51), by dividing each term by sy. If we make yox=1,

ay

bx y=t, we shall have

+ y

y groep mebep z, and y's 269=19, and yup-s6 gbp-61=xP H. Let now ap-fq=m, bp-gq=n, and we shall find p= ng— 1

abo bn-ат and 95 and by substitution + ut

A t = 0, orze itt i=0; integrating q2++pt+=C, or (bn-am) (Y* 3")*s+ (ng-m) (y3r) *-* = (agbf)C=C'.

129. Let now ġ+Pyr=aqi, P and Q being functions of x: Suppose, according to the method of Bernoulli, that y=XZ, X being some other function of x, then Xi+zX+PX ze=aQi. If we nov make 28+PXzi=o, we shall have s-Pr, and log X=-|

X
Mm

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+

y

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ag-bf'

ambn

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