16 [March G, 



If from these n 1 equations we eliminate the ml quantities 

 X 2 X 3 . . \ m , we shall obtain n m differential equations. These wiL 

 be capable of expression as exact differential equations, and wiL 

 give by integration the n m integrals before mentioned. 



The method above described admits of important applications. It 

 enables us to assign beforehand the conditions of success in the 

 application of Monge's and of similar methods to the integration of 

 partial differential equations of the second order, and even to deter- 

 mine the nature of the theoretically possible integral where its 

 actual exhibition in a finite form is impossible. It also enables us 

 to investigate by a new and perfectly rigorous method the conditions 

 of integrability of ordinary differential expressions. 



I subjoin a single result of the former of these applications. It is 

 known that the equations of the possible envelopes of any surface 



***$(*i'y> *%'*) 



in which three parameters, a, b, c, vary in subjection to two condi- 

 tions, 



/ 1 (,5,c) = 0, / 2 (M,c)=0, 

 will satisfy a partial differential equation of the form 



The application of the above method shows that, in order that this 

 equation may admit of an integral of the above species, i. e. an in- 

 tegral interpretable by the envelope of a surface in which three para- 

 meters vary in subjection to two connecting relations, the following 

 conditions are necessary and sufficient, viz. 



S 2 +4RT-4V=0, ................ (1) 



AR+ A'w=0, .................... (2) 



Am+AT=0, .................... (3) 



in which m is one of the equal roots of 



w 2 --Sm+RT-V=0, 

 and 



A d . d d . rr\d 

 A = -- +p m +T , 

 ax dz dq dp 



A ' 1* d d . T-> d 



A = m 



dy dz dp dq 

 The first only of the above three conditions appears to have 

 been assigned before (Ampere, Journal de 1'Ecole Polytechnique, 

 Cahier xviii.). 



