On the calculus of symbols.—Fourth memoir. With applications to the theory of non-linear differential equations

Abstract

In the preceding memoirs on the Calculus of Symbols, systems have een constructed for the multiplication and division of non-commutative pmbols subject to certain laws of combination ; and these systems suffice ,r linear differential equations. But when we enter upon the consideration f non-linear equations, we see at once that these methods do not apply, t becomes necessary to invent some fresh mode of calculation, and a new iotation, in order to bring non-linear functions into a condition which dmits of treatment by symbolical algebra. This is the object of the f l ­ owing memoir. Professor Boole has given, in his Treatise on Diffeiential equations,’ a method due to M. Sarrus, by which we ascertain whether a jiven non-linear function is a complete differential. This method, as will )e seen by anyone who will refer to Professor Boole s treatise, is equivalent :o finding the conditions that a non-linear function may be externally livisible by the symbol of differentiation. In the following paper I have riven a notation by which I obtain the actual expressions for these con­ ditions, and for the symbolical remainders arising in the course of the livision, and have extended my investigations to ascertaining the results )f the symbolical division of non-linear functions by linear functions of the symbol of differentiation. Let F (x, y, y lt y2, y3 , . . . y„) be any non-linear function, in which % y2, y3, . . . . y„ denote respectively the first, second, third, . . . . wth differential of y with respect to (x). Let Ur denote f d y r, i. e. the integral of a function involving x, y, y„ y2. . . . with reference to yr alone. Let V,. in like manner denote — when the differentiation is supposed dyr effected with reference to yr alone, so that Vr Ur F = F . The next definition is the most important, as it is that on which all our subsequent calculations will depend. We may suppose F differentiated (m) times with reference to y„, yn_i, or yn_2, &c., and yn, y»_i, or yn2> &c., as the case may be, afterward equated to zero. We shall denote this entire process by Z(“}, Z&& &c. The following definition is also of importance: we shall denote the ex­ pression d . d . . , T*+ y ' dy+ »’ d f + y° W ,+ ' + dyr