bo 
“I 
1900. No. 5. ON PARTIAL DIFFERENTIAL EQUATIONS, ETC. 
By now eliminating & from the first and the fourth equation, and 
multiplying the equation obtained by an indeterminate quantity u, and 
adding the result to the second equation, we obtain, a result which, 
reasoning as on page 21, may be expressed in the form 
area 
where w is determined by the equation 
FGu? + CGu + HG — AD— 0. 
It is then easy to see, reasoning as before, that the only possible 
combination of equations is the following: 
Ou Qu 
HR EE 
D 
ou AF + BG du G Qu LE 
dr Go = 
Ou ou ou 
TA; G 5 „= 
where wu is determined by the above equation of the second degree, and 
where 
(FA + BG) [AF? + G (BF + GC)| = G3 (AD — HG). 
Conversely it may be shown, in the same manner as the above, that 
a solution of any one of the systems of equations which are satisfied by 
u, should lead to the given partial differential equations of the third order. 
db. Let A=o, E+0, G +o. System (Il) (p. 16) then becomes 
Ou 
Er +G™ 
Ou\ Ou 7 du\ (ou Ou ou 
Aly a +P 15 ie, Se or ot 
