of Linear Partial Differential Equations. 27 



Now as g and g' are arbitrary, subject only to the condition 

 of being unequal, and hk are also arbitrary, subject only to 

 the condition that k must not be zero; if C"^ be not equal to 

 4<ai, we may assume gg' to be the roots of a-\-Cg-\-bg^^=0, 

 and hk such as to make the 4th and 5th terms of the above 

 equation vanish. By this method the reduced equation takes 

 the form 



This equation we may now further transform by writing 

 ^g'x+my+nz p^,. ^^. \^y which meuus we obtain 



+ {2nc + C) ^ + {cn^ + Cmn + A' I + Bhn + C'n + D)v. 



Now as /,?w, w are absolutely arbitrary, we can always so assume 

 them as to make the 3rd, ^th, and 6th terms of this equation 

 vanish. The reduced equation then takes the form 



dz* ax til/ dz 



which may be further reduced, as before, to the form (3.). 



4. But if C'^=4a6, we cannot then assume^ and ^ to be 

 the roots of the equation a-\-Qg-\-hg^=.Q ; let us therefore in 

 this case assume ^^ = 0, and g' such thato'H- C^' + 6^'^ = ; and 

 hk as before: it will be found that by this means the 2nd, 4th, 

 .'jth and 6th terms of the first transformed equation in art. (3.) 

 vanish, which therefore takes the form 



d'^u d^u .,du „, du „, du -^ / , v 



"""S^+^S^+^'s+^rf^+C'j^+D^. . (8) 



This equation we may now further reduce by the same method 

 as was employed in reducing (7.)s and it thus becomes 



d^v d% ,^ , ... dv ,-,, dv ,^ „,, dv 



+ (a/2 + cn^ + A7 + B'm + C'w + D) v. 

 Now as /, OT, n are absolutely arbitrary, we may assign such 

 values to them as to make the 3rd, 5th, and 6th terms of this 

 equation disappear; which being done, it takes the form 

 dhi d^u r>tdu 

 dx-^ dz^ dy 



But if we change the variables of this equation by writing 

 ^■=.x-\-fz.t J^ — x+J'z, it becomes 



« / . /-9X <^'^ /^ ^ ,v/^ d'^u , rio.d'^n rt.du 



