26 ALF GULDBERG, M.-N. KI. 
the five quantities 
ou ou Ou ou Mu 
| N Tee 
The actual result will be 
GAa + [uEF — AEG — AF] 8 + [REF — u FG — ED] y+ GDö+ 
+ GE (82 — ay) + GF iy? — på) + 6? (ad — fy) + AD—WEF =o, 
which, in the case here considered, G +0 reduces to the given partial 
differential equation (I).1 
The determination of a first integral of the given equation (I), in the 
case in which none of the quantities Æ, #, G vanish, is thus under certain 
conditions reduced to the integration of a system of three Znear partial 
differential equations of the first order. We shall now consider the case 
in which oze of the quantities Æ, F, G vanishes. 
Case IT. a. Let E=o, F +0, G +o. System (I) (p. 16) then 
becomes 
ou ou 
Ou\ ou Het Ou 
- = ot Ve -6f 7) = ae at 
ou ou Ou ou ou]? = eu G(s ou 
es cee or oe À = ler ae GS )a —° 
ou]? Ou ou pi ou ou 
JE å SPR De ae om ee): En dr | an — 
N NER, ‘ : 
Eliminating = from the first and the third equation, we have 
Ou på _ AF + BG du _ ul __ 
ar | or G et | 
1 It is easy to show that 
HEF — MEG — AF = BG 
AEF — u FG — ED = CG, - 
For, multiplying the first equation by å, and the second by #, we obtain 
REG + 1(BG + AF) = ku EF = AD — HG 
WFG + u (CG + ED) = å EF = AD — HG, 
