1819.] Royal Academy of Sciences. 225 



duced by the integration of these five differential equations. 

 Now the method which I propose consists in avoiding the intro- 

 duction of these constant quantities, or rather in supplying the 

 place of these constant quantities by particular values, attributed 

 to the unknown quantities y, z, u, q, r, and in integrating the 

 five differential equations, in such manner that for x = x , we 

 may have y = y Q , z = z , u = u , q = q , r = r Q ; y , z , 

 designating two new variables ; u an arbitrary function of these 

 variables themselves, similar to the arbitrary function of y and 

 ofz, which represent the value of u for x = „r ; and q , r Q , the 

 two partial derivates of u Q relative to y Q and to z . If, in the five 

 integral equations thus obtained, q and r are eliminated, there 

 will remain only three formula?, the system of which will be 

 proper to represent the general integral of the equation w^th 

 partial differences. These three formulae will contain the variable 

 quantities x, y, z, u ; the constant quantity x Q , the two new 

 variable quantities y , z , and the arbitrary function of these new 

 variable quantities represented by u of as well as its derivatives of 

 the first order relative to y and to z . It is not until the arbi- 

 trary function in question has been fixed that by eliminating the 

 new variable quantities y Q , z 9 , we can obtain the limited equation 

 which determines u in a function of or, y, z. 



Nothing hinders us from preserving in the calculation the 

 quantity p along with the variable quantities x, y, z, u, q, r, if it 

 is observed besides, that the independent variables, x,y, z, may 

 be exchanged for one another relative to the parts which they 

 act, the following rule will be obtained by the general integra- 

 tion of an equation with partial differences for three independent 

 variable quantities, and even for any number whatever of variable 

 uantities. 



Substitute, according to the ordinaiy methods, in the place of 

 the given equation with partial differences, so many differential 

 equations of the first order (minus one) as it contains variable 

 quantities, comprising therein the independent variations, the 

 unknown function, and its partial derivatives. The independent 

 variable quantities are to be treated symmetrically in the differen- 

 tial equations, one of which may be supplied by the equation 

 with given partial differences. 



This being done, integrate the differential equations in ques- 

 tion relatively to all the variable quantities that they contain, 

 setting out from certain limits which you consider as new 

 variable quantities, subjected to the same relations as the first. 

 Then, in the integral equations thus obtained, regard one of the 

 new independent variable quantities as being reduced to a con- 

 stant quantity, and the others as what ought to be eliminated, 

 you will have a system of formulae proper to represent the 



feneral integral of the equation with given partial differences, 

 hese formulae will contain only one arbitrary function, with its 

 partial divisions of the first order ; that is to say, the new variable 

 Vol. XIV. N° III. P 



