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ADDITIONS,

BY THE EDITOR, R. ADRAIN.

New Method of determining the Angle contained by the chords of two sides of a Spherical Triangle.

See prob. v. page 79 this vol.

THEOREM.

If any two sides of a Spherical Triangle be produced till the continuation of each side be half the supplement of that side, the arc of a great Circle joining the extremities of the sides thus produced will be the measure of the angle contained by the chords of those two sides.

DEMONSTRATION.

Let the two sides AB, AC of the spherical triangle ABC be produced till they meet in G, and let the supplements BG, CG, be bisected in D and E, also let the chords amв, anc of the arcs AB, AC be drawn ; and the great circular arc DE will be the measure of the rectilineal angle contained by the chords

AMB, ANC.

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Let the diameter AG be the common section of the planes of ABG, ACC, and r the centre of the sphere, from which draw the straight lines FD, FE.

Since, by hypothesis, GE is the half of oc, therefore the angle at the centre GFE is equal to the angle at the circumference GANC (theo. 49. Geom.) and therefore anc and FE, being in the same plane, are parallel: in like manner, it is shown that FD and amв are parallel, and therefore the rectilineal angles BAC and DEF, are equal, and consequently, since DE is the measure of the angle DFE, it is also the measure of the angle contained by the chords amb, anc. Q. É. D.

New method of Determining the Oscillations of a Variable Pendulum.

The principles adopted by Dr. Hutton in the solution of his 51st problem, page 541 this vol, are, in my opinion, erroneous. He supposes the number of vibrations made in a given particle of time to depend on the length of the pendulum only, without considering the accelerative tension of the thread; so that by his formula we have a finite number of vi. brations performed in a finite time by the descending weight, even when the ascending weight is infinitely small or nothing. Besides, the stating by which he finds the fluxion of the num ber of vibrations, is referred to no geometrical or mechanical principle, and appears to be nothing but a mere hypothesis.* The following is a specimen of the method by which such problems may be solved according to acknowledged principles.

PROBLEM.

If two unequal weights m and m', connected by a thread passing freely over a pulley, are suspended vertically, and exposed to the action of common gravity, it is required to investigate the number of vibrations made in a given time by the greater weight m, supposing it to descend from the point of suspension, and to make indefinitely small removals from the vertical.

SOLUTION.

Let the summit A of a vertical ABCDE be the point from which m descends, в any point in ae taken as the beginning of the plane curve вmDn described by m, which is connected with m' by the thread am. Let me be at right angles to AE, and put Ac=x, cm=y, am=r; also let , t and T be the times of the descent of m through the vertical spaces AB, AC and BC; g = 321 feet, = the measure of accelerative gravity; f = the mea. sure of the retarding force which the tension of the thread exerts on m in the direction ma, and c definitely small horizontal velocity of m at B.

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A

B

CAM

D

E the in

the vertical action of the tension on m;

fx

the true accelerative force with which

T

m is urged in a vertical direction.

Again, r : y : : f: Jy

fy

= the horizontal action on m produced

by the tension of the thread am.

Thus the whole accelera

tive forces by which m is urged in directions parallel to ≈ and

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increase x, and the latter to diminish y; and therefore by the general and well-known theorem of variable motions (See Mec. Cel. B. 1, Chap. 2), we have the two equations

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But by hypothesis, the angle mac is indefinitely small, we have

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of which the proper fluent is x=(g-f)ť: and by substituting for x the value just found, our second fluxional equation becomes

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Now when p is less than 1, let q=√-p, and in this case the

ty

correct fluent of the equation+py=0, is easily found to be

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from which equation it is manifest that as t increases y also increases, so that m never returns to the vertical, and there are no vibrations. Again, when p = 1, the correct fluent of the same fluxional equation is

2=√ hyp. log. (—).

So that in this case also, when t increases y increases, and the body m never returns to the vertical. Since in this case p 4m'

m-m

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=1, therefore 17m'=m, and therefore by this case and the preceding, there are no vibrations performed by the descending weight m when it is equal to or greater than 17 times the ascending weight m'.

VOL. II.

74

But when p is greater than, put n=✓p—, and in this case the correct equation of the fluents is

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This equation shows us that we shall have y = 0, as often as

n. hyp. log. – becomes equal to mi-circumferences: if therefore ber in the series 1, 2, 3, 4, 5,

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any complete number of se

3.1416, and N=any num. &c. we can have y=0 only

when n. hyp. log. -=x, from which we have t=s.e

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which shows the relation between the number of vibrations N and the time T in which they are performed.

Hence it is manifest that the times or durations of the several successive vibrations constitute a series in geometrical progression.

DESCRIPTIVE GEOMETRY.

CHAPTER I.

Containing the First Principles of Descriptive Geometry, with Illustrations.

DESCRIPTIVE GEOMETRY is the art of determining by constructions performed on one plane the various points of lines and surfaces which are in different planes. The principle on which this art is founded, consists in projecting the points of any line or surface on two given planes at right angles to each other. These two planes are usually denominated the horizontal and vertical planes, or the fundamental or primitive planes, or the planes of projection. In the constructions the vertical plane is supposed to have revolved about the line of their common intersection, and to be coincident with the ho rizontal plane; and it is by means of this coincidence that both the projections on the horizontal and vertical planes are effected by constructions performed on the horizontal plane.

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To illustrate this, let ABCD be the horizontal plane, and EFGH the vertical plane at right angles to it, and meeting it in

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