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stances; but there is a method at once common to algebraic equations, differential equations, and those of finite differences, which leads from particular solutions to the general: this, however, more properly belongs to the

THIRD CLASS OF EQUATIONS.

The most general form of an equation in finite differences of any order and of any degree is represented by

‘p {a, u, u, , , u, ........ u,...} = 0. Put u, - f'(y) = f(x), and wherever a enters, let its value |. # be . Y substituted, z consequently entering in a different form from ar, the transformed function may be represented by

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Differentiating relative to x, and then putting z = 0, the result is manifestly of the form

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from whence f(0) is known in terms of the indeterminate quantity y, unless F, =0 when y becomes known, and f'(0) remains the indeter

minate constant.

The successive differentiations putting x=0 after each, will determine f"(0), f"(0), and thence by Maclaurin's Theorem,

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which solution is particular, since it contains only one arbitrary constant, viz., f'(0) or y; denote this value of u, by X, and put u. = X, + v., and having found v, with another arbitrary, from the transformed equation, put v. = X, + t., X, containing two arbitraries, and by continuing this process, we shall obtain the complete solution, viz.,

R. MURPHY.

CAIU's College,
Nov. 1, 1835.

IV. Geometrical Theorems, and Formulae, particularly applicable to some Geodetical Problems. By WILLIAM WALLACE, A.M. F.R.S. Edin., F.R.A.S. Lond, Member of the Cambridge Philosophical Society, and Professor of Mathematics in the University of Edinburgh.

[Read Nov. 30, 1835.]

ART. 1. The Geometrical Theorems and Trigonometrical Formulae which are given in this paper are peculiarly applicable to the solution of some Geodetical Problems, in particular to this which follows.

“Three stations being given in position, or else the angles made by the lines which join them; also the angles which these lines subtend at a fourth station in the plane of the others; to determine the position of that fourth station.”

This problem is remarkable on account of its antiquity, and the object to which it was applied. Hipparchus made use of it to determine the position of the Moon's apogee and the radius of her epicycle, and Ptolemy actually resolved it by a trigonometrical computation in his Mathematical Syntaxis". Vieta has given a geometrical construction in his Apollonius Gallust. He had in view the solution of Hipparchus' problem; but the fiction of epicycles being now rejected, Ptolemy's application of the problem is merely an interesting fact in the history of the ancient Astronomy.

* Histoire de l'Astronomie Ancienne, par Delambre. T. II. p. 150–164.

t Vietae Opera Mathematica, p. 344. Edit. 1646.

2. In comparatively modern times an interesting application of the problem was made to geodetical measurements. In the year 1620 Snellius, when ascertaining the distance between Bergen-op-Zoom and Alcmaer, with a view to the determination of the magnitude of the Earth, employed it in finding the position of his Observatory. He assumed as given points three stations whose positions had been determined, and taking the angle which each two of them made at the Observatory, he was able to determine, by a trigonometrical computation, the distances of the stations, and thence its position*. The same problem was proposed by Richard Towneley as a chorographical problem +, and resolved trigonometrically in the Philosophical Transactions about the year 1670, by John Collins.

3. We are informed by Delambref that Lalande wishing to compute some observations of the Moon which had been made at the Military School, Paris, proposed to find the longitude of the station where the observations had been made, by observing there the angles subtended by three steeples whose positions were known. He was thus led to the same application as had been made long before by Snellius, without knowing or without thinking of his solution. Lalande's patience was exhausted by the length of the calculations, and the slips he made in performing them: he therefore referred the problem to Delambre, who gave a solution which was printed in Cagnoli's Trigonometry (First Edition), and again, but with more detail, in his own treatise Methodes Analytiques pour la Determination d'un Arc du Meridien.

Delambre's solution, which is analytical, is good, his formulae have however but little of that symmetry and simplicity which constitute

elegance in a geometrical speculation, and make it easy to be comprehended and remembered.

4. In considering the problem I have found two Theorems; from one of them a particularly simple and elegant geometrical construction is obtained, and both suggest various solutions, some lineo-angular, others trigonometrical, to this and other related problems.

* SNEllius, Erastosthenes Batavus, p. 203. + Lowthorpe's Abridgement of Phil. Trans. Vol. I. p. 120. f Histoire de l'Astronomie Moderne, T. II. p. 109.

THE OREM.

“Let AB, AC be two straight lines which meet in A; and AD, Ad other two, which make with the former equal angles BAD, CAd, these last being either both within (figs. 1, 2) or both without the angle

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BAC (fig. 3). Let the lines be such that the rectangle D.A.A.d is equal to the rectangle B.A.AC; draw lines from D and d to B and C: the triangles ADC, ABd, thus formed, are similar; and the triangles ADB, ACd are similar.”

DEMONSTRATION. Because by hypothesis AD. Ad=AB.AC, therefore AB : AD=Ad : AC. Now the angles BAD, dAC are equal by hypothesis, therefore the triangles BAD, dAC are similar.

Also because AB : Ad=AD : AC, and the angles B.A.d, DAC are equal, for they are the sums or differences of the equal angles B.AD,

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