Partial Differential Equations An Essay towards an entirely new Method of Integrating them

Abstract

THE present work, as its title indicates, contains a detailed explanation of a new method of integrating Partial Differential Equations; it is in no sense a textbook or introduction to the subject. The author's object is not to collect and describe the known methods, but to develop a new one. The principle of the method is easily explained and understood. The independent variables in the given differential equation being t, x, y, z ..., we can transform it so that the new independent variables are t, ξ, η, ζ, ..., by equations of the form D_tu = d_tu + d_xu. D_tx + ...; but the practical application of the method consists in comparing the original equation with the equation last written, and thus determining relations from which, by the elimination of φ, η, ζ ..., the integral of the original differential equation is found. The quantities t, ξ, η, ζ, ..., with the exception of the one with regard to which the differentiation is being performed, are treated as constants, and are here called quasi-constants (semi-constants we should have preferred). Mr. Earnshaw, as is apparent from the equation of transformation quoted above, adopts d when the differentiation is with regard to the old variables, and D when with regard to the new; the suffix notation for differential coefficients is also made use of. For this latter departure from custom the author in the preface offers an apology, and states that he has been warned that it “will form a serious hindrance to the acceptableness of the present work.” This fear we think is groundless; the notation is not inconvenient in such investigations as the present, as it somewhat simplifies the appearance of the equations without rendering the analysis more difficult to follow.

Partial Differential Equations. An Essay towards an entirely new Method of Integrating them.

By S. Earnshaw, M. A. (Macmillan and Co., 1871.)