1894.] Discussion of Linear Diffei'ential Equations. 207 



does not vanish. Rejecting this special case, we see that X, Y, 

 Z, T are solutions of the differential equation mentioned in 

 Art. 1. 



Writing out the equation A^y =: at full length, and making 

 use of the equations satisfied by p and q, we obtain 



,3r ,az .dx dY , dz 



aSr ab 1^ at ^ ac -^ he ^r- 



ox oy oz oz oz 



{ (dT dX\ , dX dX dZ 

 ^^^T\dy-^d^)^^'^^3-dl^'dz 



{ ,fdT dY\ ,dX dY jdZ . , ..dZ] 



( fdY dX dZ\ , dZ] ^ 



which furnishes us with four linear relations connecting the first 

 differential coefficients of X, Y, Z, T. 



5. Proceeding in the same manner as in Art. 1, we obtain the 

 result 



+ (/l . n - «1 Pn-l) ^«l (^1 + Vl^-2 + • • • + Pn-X ^n) 

 = ^(^r<+^fr,s^r^s (2), 



where the ^'s satisfy n — 1 equations of the form 

 and ^(n — 1) (w — 2) equations of the form 



«! (PrP. + PsPr) -/l, .+1 Pr "A ,+1 Ps +fr+u .« = 0. 



Further, if we write 



m = ^^u^ + ^2^*, + . . . + |^„_iii„_i, 



and A = ^ + ^,^^+...+^„_,g^, 



we have 



' — const., ^m = 0. 



\-\ 



It is therefore possible to develope a theory exactly similar to 

 that we have constructed for the case of three variables. 



