1894.] Discussion of Linear Differential Equations. 20^ 



of the four scalar solutions obtained from the matrical solution, 

 we have 



dT JdX aF> 



V dx dy J ' 



dt \ dw dy 



dT^dX_^,dZ ^^ 

 dx ' dt dy ' 



dy dt dx ' 



8F_aZ d_Z _^ 

 dx dy dt 



Another equation that may be discussed with the aid of the 

 theory as applied to three variables, is the equation 



de „ d'0 

 We have 



dt * dx'- 



aV — ty — d^ (x — pt — qy){x +pt + qy), 

 where p and q satisfy the equations 



/ = 0, ^^ = 0, pq + qp = -r,. 



iv 



Putting y = 1, we have 



aV — t = a^(x — pt — q) {x ^ yt -\- q), 



and consequently 



2 9' d ,fd d \/d d \ 



''d^-dt^'' [dx-Pdi-'^)[dx + Pdi + V- 



To adapt our theory to this case we have, therefore, to write 



u = t — px, V =1 — qx. 



Also, in this case, the four solutions will be connected by 

 the linear relations 



,dT ^ ^ dT dX ^ 



dt dx 

 VOL. VIII. PT. III. 15 



