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ting the variables, we shall obtain and taking the duents

y alr=mly +(n-m) lc;t from which we easily deduce your"-- the equation required. Prob. II. Required the curve of which the subtangent ya = a + .*

y By first separating the variables, we find that y=.

y ad + xir by integrating, that yy=(aa +4x), an equation to the hyperbola.

Prob. III. To find the curve in whicb the space APM=AMQ.

; and

, .

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By the problem we have * Sxy = fyż, bence my

or y~=:"an-n. y Prob. IV. To find the curve BM, in which the space ABMP=the arc BM mul. tiplied by a constant quantity a, or such that jyš=af ( '+y).

B
Separating the variables, we obtain
ay

; and therefore
(yy-aa)
:1
y+ (yy-aa)

is the equation required. Prob. V. To find the curve AM, in which the radius of curvature MC=MMN.

a

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If we suppose e constant, we shall have by art. (31),=

+y

en

У +yo=o. To integrate, let ;=re and we • Because from the article quoted it appears that MN=r{+;"}

while MC=(+ju)?

.

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and pa

m

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sh

m y=_P P

y P(p' +1 Therefore *.1 Y=1 P

Eyou

and (p? +1)

w (

cy)
=
Ey" y

which is the fluxional equation of the first order (

cy)
of the curve required.

If n=m, we have x=yy 100-yy), and <=cv(cc-yy), an equation to the circle. If m=2n, we have = ývy,

an equation

(C-y) to the cycloid.

Prob. VI. Required a curve BM, such that drawing through the vertex A a right line AO, making with the axis an angle of 45", we may always have the following proportion.

As the ordinate PM is to the subtangent PT, so is a given line a to OM.

By the enunciation of the problem it
appears that y:*::a:y- &, and
ar=(ym) y. Let y-r=, and we

T
shall have y
and integrating =

and
x=y-a+Ceis.

We might have found this fluent directly, by comparing the equa

A

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aand

tor

što , with those in articles 126 and 127. The least ardinate BP will be found by making =~, and then we find BD = AD=al, and the space DBMP=xy-1 yg +aal + gaat om

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Fig. 3.

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Á rocket Case of Mathematical Instruments contains tho

following Articles : 1. A pair of PLAIN COMPAsses. 2. A pair of Drawing ComPASSES. with its several parts. 3. A Drawing Pen and Pointer. 4. A PROTRACTOR, in form of a SEMICIRCLE, or, sometimes, of a PARALLELOGRAM. 5. A PARALLEL RULER. 6. A PLAIN SCALE. 7. . SECTOR : besides the Black LEAD PENCIL for drawing Lines.

1.

OF THE PLAIN COMPASSES, FIG. 1. THE use of the Common or Plain Compasses is, 1. To draw a Blanik Line AB, by the edge of a ruler, through any given point or points CD, &c. 2. Take any extent or length between the points of the Compasses, and to set it off, or apply it successively upon any line, as from C to D, fig. 2. 3. To take any proposed line CD between the points, and by applying it to the proper Scale, to find its length. 4. To set off Equal Distances upon a given line, by making a dot with the point at each, through which to draw parallel lines. 5. To draw any blank circle, intersecting Arches, &c. 6. To lay off an angle of a given quantity upon an arch of a circle froni the line of chords, &c. 7. To measure any arch, or angle, upon the chords, &c. 8. To construct any proposed figure, in plotting or making plans, &c. by setting off the quantity of the sides and angles from proper Scales. In short, the use of the Compasses occurs in every branch of pracical mathematics, as we shall see more particularly hereafter.

II. OF THE DRAWING COMPASSES.

These Compasses are chiefly designed for drawing circles and circular arches ; and it is often necessary they should be drawn with different materials, and, therefore, this pair of compasses has, in one of its legs, a triangular Socket and Screw, to receive and fasten the following parts or points for that purpose, viz. 1. A Steel point, which, being fixed in the socket, makes the Compasses then but a plain pair, and has all the same uses as just now described in drawing

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