ÆäÀÌÁö À̹ÌÁö
PDF
ePub
[blocks in formation]

and consequently whether we admit ., ., .... to be comprehended in or not the whole expression (35) sinks to an arbitrary function of sin 2, that is to ₫ sin

n

2 kλ z

n

,་,,

And the same coincidence might easily be shown to be true of any other apparently different integral, which from the nature of (34) must contain circular or equivalent periodic functions, so that (34) has really but one integral, involved, however, in an arbitrary function. This is likewise confirmed by a very simple, direct, and general method that has occurred to me of integrating equations of differences of, I believe, all orders and degrees; and which method is applicable to the direct numerical resolution of algebraic equations of all degrees as well as to other purposes, of which I may say more hereafter. Hence (25) or (26) does not admit of variety from variety of integral, but being obtained directly and without any limiting conditions from the integration, it must possess an equal degree of generality with the integral itself, that is, the integral being of the most general kind, the functional solution must be so too, or in other words complete. A complete solution, therefore, of 4" x = x has only one arbitrary function.

-1

[ocr errors]

A direct proof that our (25) is the complete solution may be given thus. Let a 4-a denote (25), or any other solution wherein a is the particular form of a when x = x. Suppose fx is also any solution whatever that will fulfil the condition fxx. Put ƒ3 x = Put fx = a - x, and changing x into x, 4 we have fa a x, in which is the function to be determined. Now because f and a are periodics of the same order, this equation is always possible, whatever be the forms of ƒ and a; and indeed I have in another place given general solutions of this very problem. It is, therefore, evident, that such a form can be given to the arbitrary function in (25), that this solution shall coincide with any other solution whatever. Consequently (25) comprehends every solution, and is, therefore, the complete solution.

This I believe is the first direct and legitimate demonstration that has been given of the completeness of a functional solution.

Hence it follows, that the arbitrary constants a,, b, c, in (21) form a part of the arbitrary function in (25). This is further confirmed by the assumption with which we set out at the commencement of § 3, which, being arbitrary, of course gives an arbitrary form to the functional root. Arbitrary functions, therefore, of whatever kind or quantity they may be, substituted for these constants, merely clog the solution without at all contributing to its generality; since these functions, and the

1

consequences that can in any way flow from them, are naturally comprehended in the arbitrary function of the direct solution.

I cannot refrain here, while on the subject of periodic functions, from mentioning two curious cases of periodic solutions which are perfectly successful when the operations are merely indicated but not performed, and yet fail when developed; though the developed and undeveloped values are the same. The first I shall mention was noticed in elucidating some difficulties in this calculus to my promising friend and pupil Mr. Mervyn, Crawford, and the next occurred while considering the source of what Mr. Babbage denominates "a very difficult subject." It was mentioned to him in a letter dated March 22, 1824.

Let be a periodic of the second order whose solution is evidently

F (x, 4 x) = 0,

where the form of F is to be determined. Substitute for x, and it becomes

F {4 x, &2 x} = F {4 x, x} = 0 = F {x, & x} Fis, therefore, symmetrical with respect to x and ↓ x. quently a solution is,

in which n is unlimited, or

4 x" + x” = 0,

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

Now in all cases the first of these values

conditions of the question; for 42 x =
"x"= x. But in the second value x W

↓2 x = x. (

[ocr errors]
[blocks in formation]

Conse

[blocks in formation]
[ocr errors]

which if n be of the form, 2 p becomes 42 x = x —1, an expression that cannot, with any integral value at least to P, be = x. It may be asked, in what this unexpected anomaly consists? The answer, I conceive, is obvious, if we seek it from the nature of the functions in question. Periodic functions are algebraic expressions whose property of periodicity depends not on the value but the form of the expression. If, therefore, the value be the same but the form be changed, the expression may no longer be periodic. Thus it happens in the above instance, the values of the two expressions are the same, but the forms different-the negative sign in the one case being a mere sign or subtraction, and in the other a factor.

This reasoning will appear still clearer in the following case. Suppose we have the equation

[blocks in formation]

1

c being a constant. By changing a into, it appears when

[ocr errors]

n = 1, that c can only be + 1 or 1. If, however, Laplace's method of differences be followed, we have for the solution of (37)

log2 A. - log r

+ x = A1

log n

(38)

A, A, being the constants of integration. If now we change x into, the solution becomes, putting p for the former exponent,

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

which, therefore, satisfies the conditions of equality of the question without any apparent limitation to c, that is, without the necessary limitation the question requires. A part only, therefore, of the conditions of the question are satisfied; and this arises from the periodicity of the exponent being destroyed by the development of p. If we consider that by changing x into twice successively, p returns into itself, and suppose p by one such change to become q, we may easily find that

[ocr errors]
[ocr errors]

And in the same way if n = 1, we should have c = VI, provided the value of c be sought by the non-development of the function.

These illustrations will, I hope, obviate the difficulty noticed by Mr. Babbage in Phil. Trans. 1817, without having recourse to the ingenious but, I presume, controvertible idea of the function, having simultaneously different values in different parts of the same equation. The same may be said of Mr. Herschel's views, p. 120, vol. ii. of the Examples.

As this is a subject of considerable importance in the theory of functions, I shall here briefly notice another more general instance of my observation; namely, that the same expression may be periodic or non-periodic according to its form. In our solution (25), k and are perfectly arbitrary. Take then r = sin -I X, and we have

2 kv

n

2 kvλ

n

4° x = sin ̄1 sin {2 **^ + sin1sin x} +x....(39) the former value of which is periodic for every value of n, provided sin and sin-1 act separately and distinctly, but the latter is not. I am aware it will be contended that the arithmetical values of ↓ x in (39) when differently developed, are not necessarily the same. This may be true in the present case, but it

will not hold in (36) or (38), and, therefore, does not militate against the general truth of my position, that the same expression may be periodic or non-periodic according to the form under which it is put. JOHN HERAPATH.

ARTICLE IV.

Instructions respecting Paratonnerres, or Conductors of Lightning. Extracted from the Report of M. Gay-Lussac, in the name of a Commission appointed by the Royal Academy of Sciences of Paris.* (With a Plate.)

THE principal object of the report (which was drawn up at the request of the Minister of the Interior), is to direct workmen in the construction and mode of fixing conductors on buildings, &c. It is divided into two parts, one theoretical, the other practical.

Theoretical Part.

Principles respecting the Action of Lightning, or Electric Matter, and of Conductors.

Lightning is the sudden passage of electric matter through the air, with the evolution of great light, from clouds highly charged with that fluid; its velocity is immense, far surpassing that of a ball at the moment it leaves the cannon, and is known to be at the rate of about 1950 feet per second of time.

The electric matter penetrates bodies, and traverses their substance, but with very unequal velocities; through some, which are therefore called conductors, it passes with great rapidity; such are well burnt charcoal and water; vegetables, animals, and the earth, in consequence of the moisture they are impregnated with, and saline solutions; but, above all, metals afford the readiest passage to the electric fluid. A cylinder of iron, for instance, is a better conductor than an equal cylinder of water saturated with sea salt, in the ratio of at least 100000 : 1, and the latter conducts a thousand times better than pure water.

Non-conductors, or insulating bodies, oppose great resistance to the passage of electricity through their substance; such are glass, sulphur, the resins, and oils; the earth, stones, and bricks, when dry; air and aeriform fluids.

No bodies, however, are such perfect conductors of electricity as not to oppose some resistance; which, being repeated in every portion of the conductor, increases with its length, and may exceed that which would be offered by a worse but shorter

*From the Annales de Chimie.

conductor. Conductors of small diameter also conduct worse than those of larger.

The electric particles are mutually repulsive, and consequently tend to separate and disperse themselves through space. They have no affinity for bodies, they determine only to their surfaces, where they are retained solely by the pressure of the atmosphere, against which, they in their turn exert a pressure proportionate at every point to the square of their number. When the latter. pressure exceeds the first, the electric matter escapes into the air in an invisible stream, or in the form of a luminous line, commonly called the electric spark.

The stratum of electric matter on the surface of a conductor is not of equal density at every point of its surface, except it be a sphere. On an ellipsoid the density is greater at the extremity of the great axis than on the equator, in the ratio of the great axis to the smaller; at the point of a cone it is infinite. In general, on a body of any form, the density of the electric matter, and consequently its pressure on the air, is greater on the sharpest or most curved parts, than on those that are flat or

round.

The electric matter tends always to spread itself over conductors, and to assume a state of equilibrium in them, and becomes divided amongst them in proportion to their form, and principally to their extent of surface. Hence, if a body that is charged with the fluid be in communication with the immense surface of the earth, it will retain no sensible portion of it. All that is necessary, therefore, to deprive a conductor of its electricity, is to connect it with the moist ground.

זי

Of several conductors of very unequal powers the electric fluid will always choose the most perfect; but if their differences be small, it will be divided amongst them in proportion to their capacity for receiving it.

A Paratonnerre * is a conductor which the electric matter of the lightning prefers to the surrounding bodies, in order to reach the ground, and expand itself through it: it commonly consists of a bar of iron elevated on the buildings it is intended to protect, and descends, without any divisions or breaks in its length, into water or a moist ground. An intimate connexion of the paratonnerre with the ground is necessary, in order that it may instantly transmit the lightning as it receives it, and thus defend the surrounding objects from its attacks. When lightning strikes the surface of the ground, for want of a good conductor it does not spread over it, but penetrates below it till it meets with a sufficient number of channels to carry it completely off.

*I adopt the French term, as we have none in our language to express in one word, a conductor of lightning, meaning thereby not merely the metallic rod, but the whole apparatus complete. At least we may as well use it as parasol, parachute, paraboue, &c.-Tr.

« ÀÌÀü°è¼Ó »