DIFFERENTIAL EQUATIONS OF THE SECOND ORDER. 13 



after these substitutions are made, we have 



A' := A& 5 + UU 

 H' = 2Ag^ + H 



+ G (&,.- + &,) + F 

 and so on ; so that, if 



u (x, t/, z) = u (at', y, s), 



we have 



' 



A/ i + H ' T + = A r + H v - + 



\ 3f / of 3ij \ 9.'- / oa; o// 



on substitution and collection of terms. The equation may therefore be called the 

 characteristic, invariant of the original differential equation. 



8. A method for integrating partial differential equations of the second order, 

 when they possess an intermediary integral, has been given by me elsewhere ; its 

 aim is the actual derivation of the intermediary integral which, being of the first 

 order, can be regarded as soluble. The preceding method makes no assumption as 

 to the existence of an intermediary integral, and indeed is entirely independent of 

 that existence ; so that it can be applied not merely to that former class, but also 

 to equations that do not satisfy the preliminary conditions for the possession of an 

 intermediary integral. 



SECTION II. 

 Equations having a Resoluble Characteristic Invariant. 



9. As a first example (which, it will be seen, possesses intermediary integrals), 



consider the equation 



b=j-g+h. 

 The characteristic equation is 



pq + <f p + q = 0, 

 which can be resolved into the two equations 



The other three equations, deduced as in 3, 4, are easily found to be 



tlh , dh . dy , \ dy r 



1_ p- ^~ (P 1 *) T~ ^ 



dx dy dx dy 



dx dy do: dy 



df df , dc , x 'fc 



~"4-- h K(P "" <7 ~" U T" = *- 



dx dy dx ay 



