224 Proceedings of Philosophical Societies. [Sept. 



sition that may be actually observed between many objects, 

 which appears to us, he says, to be the highest point of abstrac- 

 tion and generalization to which science is permitted to be 

 carried. 



An Essay upon the Integration of a particular Class of differ- 

 ential Equations, and An Essay upon the Integration of Equations 

 with partial Differences of the first Order, and with any Number 

 of Variables, by M. Cauchy. 



Note by the author concerning the latter of these two memoirs, 

 Jan. 27, 1818. 



There is not at present any treatise upon the integral calculus 

 in which a method has been given of completely integrating 

 equations with partial differences of the first order, whatever 

 may be the number of independent variable quantities. Having 

 attended for several months to that subject, 1 was lucky enough 

 to obtain a general method for this purpose. But, after I had 

 finished my work, I learned that M. Pfaff, a German mathema- 

 tician, had obtained the integrals of the above-mentioned 

 equations. As this is one of the most important questions of the 

 integral calculus, and as the method of M. Pfaff is different from 

 my own, I think that an abridged analysis of the last may be 

 interesting to mathematicians. I shall, therefore, explain it ; 

 and in order to facilitate the explanation, shall profit by some 

 remarks made by M. Coriolis, civil engineer, and of some other 

 remarks which have since struck me. When thus simplified, 

 the method which I have used appears to me to furnish the 

 simplest solutions that can be given of the proposed question. 

 The following considerations will allow a judgment to be made 

 of it. 



In order to have some fixed ideas on the subject, let us suppose 

 the equation with partial differences that is proposed contains, 

 along with the three independent variables, x, y, z, an unknown 

 function, u, of these three variables, and the partial derivatives, 

 p, q, r, of the function, u, in respect to the same variables. 



In order to determine exactly the value of u, it is not suffi- 

 cient to know that the given equation must be verified with 

 partial differences. It will be moreover necessary to add some 

 condition, for example, to subject the function, u, to receive for 

 a given value, x Q of the variable quantity, x, a certain value of a 

 function of the variable quantities y and z. The function of y 

 and z here meant, which may be chosen at pleasure, is the only 

 arbitrary function which ought to be contained in the general 

 integral of the equation with partial differences. It is, in other 

 respects, easy, by means of principles which are already knowo. 

 to reduce the integration of this equation with partial differences 

 to the integration of five differential equations between the six 

 quantities, x, y, z, u, q, r, considered as functions of a single 

 variable ; and the whole difficulty is reduced to knowing what 

 must be done with the five arbitrary constant quantities intro- 



