accomplished in a few lines. To any person acquainted with the notation employed in the doctrine of functions the question is comprehended at a single glance; yet if we apply to it the rules discovered in Prob. 1 and 2, we shall find that if it were written out at length, x and y would each be repeated 512 times, and would occur 1023 times; so that the whole expression would consist of 2047 letters, and it may be added, that if it were so developed it would require a much longer time merely to comprehend the enunciation of the problem than it would to understand and solve it in its contracted form. Feb 26, 1820. CHARLES BABBAGE. VI. On the Reduction of certain Classes of Functional Equations to Equations of Finite Differences. By J. F. W. HERSCHEL, M. A. FELLOW OF THE ROYAL SOCIETIES OF LONDON AND EDINBURGH, AND OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY. [Read March 6, 1820.] THE reduction of functional equations to those of finite differences was the means by which their resolution was originally attempted and though applicable only to particular classes of them, affords very ready solutions in those instances in which it can be practised, and these solutions have the advantage in general of indicating at the same time the number and nature of the arbitrary functions they involve. I propose therefore in the following pages to apply this method to a class of pretty considerable extent, in which the function to be determined is one of two or more variables, but where the condition expressed by the equation assigns to one of them a value or values dependent in a known manner on that of the other. Mr. Babbage, in the second part of his paper on Functional Equations (Phil. Trans. 1816.) has noticed equations of this nature, and has given solutions of certain cases of the higher orders. With these I shall not at present concern myself, the method which I propose to exhibit extending only to equations of the first order, of which it affords the complete solution in all cases. To proceed regularly, it will be necessary to premise the following Problems, the object of which is to determine the nature of the arbitrary functions we shall be obliged to introduce. PROB. I. Required the most general form of a function (x, y) of two variables x, y, which remains the same whether P or Q, (two given functions of x) be substituted for y, or to resolve the equation (x, P) = √(x, Q). Since the substitution of P for y renders (x, y) a certain function (fx) of x, it is evident that in its general state (x, y) can only differ from fx by a quantity which vanishes when y becomes P, and of course (x, y) − f(x) must necessarily have y - P for a factor. Again, since y(x, y) reduces itself to the same function f(x) by the substitution of Q for y, the quantity (x, y) − f(x) must in like manner have y-Q for a factor, and we must therefore have Let this be substituted in the equation (x, P) = y (x, Q), and the whole vanishes. This value of y(x, y) therefore satisfies the equation independent of any particular forms of the functions ƒ and X, which therefore remain arbitrary. PROB. II. Required y(x, y) a function of x and y, which retains the same value when each of the functions P, Q, R, &c. is severally substituted for y; P, Q, R, &c. being given functions of x. By reasoning exactly similar it will appear that f(x) + (y−P) (y− Q) (y - R), &c. x (x, y), is the most general form of the function sought, f(x) and x(x, y) being arbitrary functions, the former of x, and the latter of x and y; observing however that x (x, y) must not be so taken as to become infinite of the first or any higher order by the substitution of any of the quantities P, Q, R, &c. for y. PROB. III. Required the most general form of a function which shall reduce itself to a given function Q when y becomes equal to another given function P, or to resolve the equation, +(x, P) = Q. Since (x, y) becomes Q when y becomes P, it is evident that must have y - P for a factor, and that (x, y) Q and x is shewn as before to denote an arbitrary function provided only it do not become infinite by putting P for y. These cases being premised, we shall now proceed to the solution of the equations we proposed to consider. PROB. IV. To begin with a simple case, let the equation proposed be h being a quantity independent on x or y, and therefore to be regarded as an arbitrary constant in the expression of ↓ (x, y). This supposition being made, we have ↓ (x, x) = p(x, h + 1); † (x, 0) = p(x, h), so that our proposed equation becomes p(x, h + 1) – p(x, h) = a. Now h being independent on x and y, this may be regarded as an equation of differences in which the auxiliary letter h is the independent variable, and its integration immediately gives p(x, h) = ah + C, in which C may be any function which does not change when h changes to h+1, that is, which retains the same value whether we put æ or o for y: now we have already seen, Prob. 1, that the most general form of such a function is Hence if we substitute for C its value, and suppose the constant ah included in the arbitrary function fx, we get finally, ¥ (x, y) = a. 2 +ƒ (x) +y (Y−x). x(x, y). Some objection may be raised against the generality, or even the legitimacy, of the above solution, on the ground that in assuming ¥ (x, y) = p(x, h+2) we in fact limit the possible forms of the y function ; for although it be true that by the combination of the elements x, h + with each other and with constants it is possible to form any assigned function of x and y, yet to render this assertion general, the quantity h must be admitted as one of those constants, and therefore our assumption should have taken notice of this circumstance, and have stood thus, ¥ (x, y) = $( Now this will very materially affect the process which follows, as the equation will then become |
