40 M. Falk, On the Integration of partial diff. Equations etc. 



Ay — Lz,^ 1 = + l/rZ7. ^ + y , Zr,, = ± V LÜ . y ^- ^ . 



Now taking- out the values of 2^2,0 ^.nd rj 1 and substituting in 

 ^^1.0 = '■i,(,dx -\- Zi-idy, this equation will become integrable, if these 

 values be taken the one for the upper and the other for the lower sign of 

 the surd. Therefore these values will satisfy the equations 



Ay — /.-,1 = + yXtJ. X -\- y 



and 



L:,.o= +VLÜ.y + è. 



Now multiplying these equations by — dy and dx respectively 

 and adding, we get 



'1 

 2 



whence by integration with respect to x 



Xci,o = +xyyLU-\- a\-+ Ix — yy + t, 



-^^y^/7-rr , A^^y 



Lz = + -f yZU + A -g- + «^ "2 — y*'.V + ta; + <p(^/), 



cp being an arbitrary function. That this mixed solution is particular, is 

 shewn by § 1. 



We do not suppose, that we have here overcome all the difficulties, 

 which exist in the delicate problem of obtaining a theory of partial diffe- 

 rential equations of the form treated in this memoir. As to our method of 

 deduction, it is essentially the same as that, which Boole in his Treatise 

 on differential equations applies to Monge's linear partial differential equa- 

 tion of the second order: 



Rr ^ Ss ~ir Tt = V. 



We hope in a future memoir to extend the theory, now given, to 

 any number of independent variables. 



