Partial Differential Equations. An Essay Towards an Entirely New Method of Integrating Them

Partial differential equations – is an essay which was published in 1871 and written by Samuel Earnshaw, an English clergyman as well as physicist and mathematician. He is known for his contributions to theoretical physics, in particular "Earnshaw's theorem". The author calls his work an essay because as a treatise it would be very incomplete, since many parts of the general topic are not mentioned in these pages. For information on these parts, he should refer the reader to Carmichael, Boole and Gregory, with whose wonderful and valuable works he tried to interfere as little as possible. In fact, he only consulted them for illustrative examples that would fit within the scope of his essay. He realizes that he has clearly intervened in the integration of the equations of one independent variable; but this arose due to the peculiarities of the author's system, with the help of which each equation, no matter how many of its independent variables, is reduced to an equivalent equation with only one independent variable. Therefore, it was impossible to separate the two branches of differential equations. The system of integration proposed here came to the author's mind several years before the writing of this essay, but his professional studies did not leave him leisure to understand the general idea in its details. And, since his goal is only to make his Method fully comprehensible, and not to demonstrate integrals, for the sake of brevity, he did not always go to the last steps of integration when he realized that the method became comprehensible. Therefore, he hastens to assure the reader that all stages of research obey one general principle and, although to some extent new, in fact are not difficult, but, on the contrary, easy when the eye is accustomed to new designations. And, moreover, he harbors the hope that the results of the integrations (many of which are much more general than in the form in which they were presented in previous Treatises) will reimburse the reader for any additional problems he may find in progressing through that, which at first may seem to him a forest of active symbols. There are many equations in this essay that are easily integrated in finite terms that, as far as the author knows, have never been combined in finite terms before. Three more important partial differential equations could hardly be cited as examples, and the finite integral of each of them is obtained in this essay without any difficulty.