24 ALF GULDBERG. M.-N. Kl. 
Supposing w to be thus determined, the equation in question 
becomes 
CCE fre 
If u, and u, be the values of u, we have 
4% - (5) t nr | pce -ga|=" 
acer] fed +) 22 
and these two equations are evidently together equivalent to the second 
and fourth equations of system (II). 
After the considerations expounded above, we may substitute for 
system (II) the following equivalent: 
au ou ou 
ee 
redd EC EEE 
when u and å are determined by the two equations (u) and (A). 
Now these equations can only be simultaneously satisfied by equating 
to zero one factor in the first member of each; and the various combina- 
tions which thus become possible give rise to different systems of linear 
partial differential equations of the first order. 
Remembering that all systems which lead to = 8) (ot = = 0, must” 
be rejected, an easy discussion shows that the only possible systems are 
