صور الصفحة
PDF
النشر الإلكتروني
[blocks in formation]

AD.= F. + (F2 + 2F). B, +3 F2. C1,
&c. = &c.

whence integrating, so that each integral may vanish when x = 0, as has been proved to be necessary, we have

[blocks in formation]

2

and v. = v。 + B ̧v2 + С ̧v3 + D.v.*, &c. is completely known.

[merged small][merged small][merged small][ocr errors][ocr errors][merged small][merged small][merged small][merged small][merged small][ocr errors][ocr errors][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small]

Before leaving this class of equations, we may remark a curious relation

[merged small][ocr errors]

which is such that the solution of one leads to that of the other, for if we put f(u) = v1, we have

Ux+1 = p(v.);
•f (u) = f¢ (0)

or v2+1 =ƒ$(v2);

if then we determine u, so that v1 =ƒ(u), v, will be readily found from u.

SECOND CLASS OF EQUATIONS.

Given {u, u x + 1 U x + 2 • • . Uz + n } = 0.

Put u1 =A1+ By* + c1 (By2)2 +c2(Bya)3 &c.

I

[blocks in formation]

and = ${41, A2, Aз......An+1},

and ultimately make A1 = A2 = A3 = ...... An+1•

Substituting in the proposed equation we have

• + г1. By* + г1⁄2. (By2)2 +г3. (By*)3 + &c. = 0,

1.

2.

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][ocr errors][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]

the equation

1

2

= 0 (putting 41 = A2 = ... ...A+1) determines A1,

T1 = 0 will give n different values of y, any of which may be used,

[merged small][ocr errors][merged small][ocr errors][merged small][merged small][merged small][merged small]

Now in linear equations the sum of the particular solutions gives the complete integral, but this is not generally the case in other in

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

[blocks in formation]

Put u, =ƒ(y3) =ƒ(≈), and wherever a enters, let its value

[blocks in formation]

substituted, consequently entering in a different form from x, the transformed function may be represented by

F{≈, ƒ(≈), ƒ(y≈), ƒ(y3≈)..............ƒ (y"≈)} = 0.

Putting ≈ 0, F{0, ƒ(0), ƒ(0).........ƒ(0)}

= 0,

from whence f(0) is known.

Differentiating relative to %, and then putting ≈ 0, the result is manifestly of the form

[blocks in formation]

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 indeterminate constant.

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

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors]

1

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 u1 = X1 + v2, and having found v. with another arbitrary, from the transformed equation, put v=X2+ t2, X2 containing two arbitraries, and by continuing this process, we shall obtain the complete solution, viz.,

2

2

u1 = X1 + X2 + X3 + ...... + X1.

2

CAIUS COLLEGE,

Nov. 1, 1835.

R. MURPHY.

IV. Geometrical Theorems, and Formulæ, 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 Formulæ 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.
+Vieta Opera Mathematica, p. 344. Edit. 1646.

« السابقةمتابعة »