of Linear Partial Differential Equations. 25 



d^u (Pu du ,,. 



dxdi/ dz^ d. 

 d'^u du 



= a 



(4.) 



dxdi/ dz 



the value of a being either zero or unity. 



I have not been able to discover that these are capable of 

 further simplification ; for which reason in the title of this 

 communication I have called them fundamental forms. If you 

 think this subject worthy of a place in your Journal, 1 will, in 

 a future communication, add to it a few remarks on the inte- 

 gration of the equations here considered. 



1. To transform 



d^u ^ J d^u p d'^u .,du , y>/^" , ]^ _^ 

 daf-^ "^ </?/^ dxdit/ dx dy ' 



the general partial differential equation of the second order 

 with constant coefficients. 



Changing the variables by means of the assumptions 



we have 



+ (2a + 2hg^ + C .gTI) ^ -f (A' + B'^0 % 



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

 of being unequal, we may assume them to be the roots of 

 a-|-C^-|-ig2=0, unless C^ = 4a6. By this assumption the 

 first and second terms of the last equation will disappear, and 

 therefore the equation will be reduced to the form 



^ d^u ..du r„du ^ , ^ . 



0=^0^-^ +A'-^ +B'-j- -\~Du. . . (5.) 

 dxdy dx dy 



Let us transform this equation by assuming u—ve^'''^'"^; by 

 this means it becomes 



4(C/w + A7+B'm + D)T;; 



and as I and m are arbitrary it is always allowable to assume 

 C7W + A' = 0, C/+B=^0, by which it is reduced to the form 



„ dhi ,^ 

 = C-j— J- -f D?<; 

 dxdy 



