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We can easily find the elements of this conic section. angle P'PS be denoted by , then evidently

h = rusing,

which, substituted in (5), gives the value of p = a(1

Thus let the

e2); next a and

, which we recall stand for ecoso and esine, are given by (2) and (3).

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consequently the greater axis, and the species of conic section described, are independent of . We have an ellipse, a parabola, or a hyperbola,

according as 2 is less, equal to, or greater than 2

M+ m

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which may evidently be taken as a general expression for the square of the velocity, if r denote the general radius-vector.

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Thus, in different orbits, the areolar velocities are as the square-roots of the parameters, and as the square-roots of the sums of the masses. In an elliptic orbit, if T denote the time of revolution, the double of the area of the whole ellipse

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Thus the theorem that, provided the sum of the masses remain the

same, the squares of the periodic times in different orbits are as the cubes of the greater axes.

The mean angular velocity is usually denoted by 1, thus,

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It is customary with astronomers to assume the earth's mean distance from the sun as the linear unit. If M and m are the masses severally of the sun and earth, and m', a' and ' belonging to another planet are introduced, the mean distance of the last is given by the equation

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To complete the subject it is necessary to notice a particular case of the problem, viz., when & = 0. Here the motion is in a right line, and from (6) it appears the velocity is infinite when the body arrives at S. As the existence of another body here ought not to be considered, at least in a mathematical sense, as an obstacle to its further motion, it is plain the body will pass beyond and move in the same right line until its velocity is reduced to zero, when it will return on its path, which will thus be a portion of a right line of which S is the middle point. This cannot be considered as a degenerate form of a conic section of which S is the focus. For when an ellipse is varied by augmenting the eccentricity but maintaining the greater axis constant, at the point the first has attained the limit unity, the ellipse has degenerated into two equal portions of right lines overlapping each other and having their extremities on one side in the point S. Hence this case must be regarded as a singular solution. However most of the properties of motion can be deduced from those of elliptic motion. Thus, if the length of the whole path be denoted by 4a, the duration of an oscillation will

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Whence we gather that the time, in which a planet, at rest at its mean distance, would fall to the sun, is found by dividing its periodic time by 41/2.

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given by Matthew Collins, on page 278 of Mathematical Monthly, Vol. 1, who states that he proposed it in an old number of the "Educational Times," but no geometrical demonstation had yet appeared. He then gives a solution of a particular case, only, namely, that in which one of the three circles that thouch each other becomes infinite.

The following method pre-supposes some knowledge of the properties of Centers of Similitude, of radical axes and of circles in contact, and it is based on a beautiful theorem, remarkable for its generality, known to the ancients under the name of "The Arbelos," or "The Shoemaker's Knife," which is enunciated by Mr. Collins, in the paper referred to, by means of a figure, as follows:

"If the semicircles on the diameters AB, AC, touching each other at A, be both touched by the circles whose centers are O and O'. which touch each other at L; demit OF, O'H perpendiculars on AB, then if OFn times the diameter of O, O'H = (n + 1) times the diameter

of O', O' being nearer io A than O is."

The elegant geometrical proof there given, holds true, if the two original semicircles touched each other externally at A.

Let the circle whose center is O", touch the circle O' and the two original semicircles, O"" touch O" and the semicircles, and so on in the same order, N being the center of the nth circle from O; it is required to find the relation connecting the quantities T, R and r, T being the

common tangent, and R, r, the radii of the circles whose centers are O and N.

This may be easily accomplished by means of the Arbelos, thus:
Putting OF=m times 2R, then NW = (m + n).2r.

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Now produce ON until it intersects the common tangent, or radical axis of the original semicircles in P; hence by a well-kdown theorem, namely, “If each of two circles touch (in the same way) another pair of circles, the center of similitude of either pair lies upon the radical axis of the other pair," P is the external center of similitude of the circles O and N, and

PO R
PN r

=

(3). Also S, S', being two anti-homologous points, (see Chauvenet's, p. 360, or any treatise on the Modern Geometry) on the circles O and N, we have the equation

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PA(Rr) NW.ROF.r2uRr from (1).

PA=

=

212Rr Rr

By substituting in eq. (5), this value of PA, and the

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which reduces to T = 2n1/Rr, the relation required; or putting R d1, r = d, we get Tny d1d

=

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If n = 2, we have the original theorem proposed by Mr. Collins.

The same method will apply, if the two original semicircles touched each other externally at A.

If we consider T the transverse, instead of the direct common tangent of O and N, then (2) becomes

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Three circles whose radii are a, b, c, touch each other externally. Within the space enclosed by them a circle is drawn tangent to the three circles, and within this circle three circles are drawn tangent to each other and to the three given circles. Calling the radii of these three circles x,, y, z,, we may determine three other circles, radii x2, Y2, 2, touching each other and the second set of circles in a similar way; and so on. Find the radii, x, y, z, of the th set of inscribed

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