10 ALF GULDBERG. M.-N. KI. 
Now to satisfy these equations simultaneously, it is necessary that we 
should equate to zero one linear factor from each of the first two equa- 
tions, and combine these with the third equation. 
If we equate to zero the first linear factors, we thus have 
ou I Ou , Ou 
ni MO ay EE ls 
au I 
u tas? 
Ou\ 2u Ou\ ou Ou du 
rer 
from the first two of which equations we derive 
Ou ou 
Ay Å» or = u = 0. 
Subitituting this value for En in the third equation, and supposing 
ou 
SF + 0, we find that 
ou ou ou 
hå A EF +2() — He =o. 
We thus have the following system of linear partial differential 
equations : 
2 Ou Ou , Ou 
Ay 5 tA 5, Hy =? 
2 Ou Ou ou 
An thx tH =? (12) 
ou ou Qu 
hjåg Å PRE Eg 
That this system is relevant to the solution of the problem under 
consideration, may be shown by eliminating from it, by means of the 
equation 
ou ou Ou Ou 
Bb eh rare] 
ou ou ou ou 
G)+ +840 
the quantities 3 
au Ou\ Ou Ou Ou 
ax)" \ay)’ ar’ ds’ à 
