6 ALF GULDBERG. M.-N. Kl. 
contain 9%, y?, ay, Bd, ad and By. The equation will in fact assume 
the form 
Aa + BB + Cy+ Då + E(®— ay) + Fy? — på) + 
+G(ab—py)+H=0 (5) 
in which 
_ Du; v) _ Du, v) __ D(u, v) 
Dr AN heel Sy GT 2)" 
Now this equation assumes the form (3), when the conditions (a) are 
satisfied — and only then. 
A consequence, which is important for the investigations in the following 
section, may here be noted, namely, that it would be useless to seek 
a first integral of the form «= /(v) for any partial differential equation 
of the third order, which is not of the form (5). 
The relations (a), as we see, are the well-known conditions, that # 
and v, considered as functions of 7, s and Z, should not be independent. 
2. Proposition. If the equation 
Aa + Bg + G+ Di+ H=o (3) 
admits of a first integral of the form u=f(v), then u and v, const- 
dered as functions of x, y, 2, p, 9, % 5, t, will each satisfy three partial 
differential equations of the form 
eu 2 Qu ou Qu ou 
er DER MET, 
as ar aar 
Qu\ ou Ou\ du ou Ou 
4(z) a +2() ap by BE 
; ; ou ou ou 
in which and a stand for ? = + pe 35 ur 3 + 32 sp and 5, =: 
3 2 
A u ou du Ou Ê 2 
+ Pie = ED ‘= “a io 7 respectively. 
By the last proposition z and v must satisfy the conditions (a) which 
are expressible in the forms 
au ov ou av ou ov - 
ag es re 0 BEE; 
Qu m’ Ou m’ On m 
ar Ss 95 or aA 
