> 
8 ALF GULDBERG. M.-N. Kl. 
If we here substitute for » and / their values from the two first 
equations (b), and set dr = adx + Bdy, ds = Bdx + ydy, dt — ydx + ddt, 
and remember that «=a is to be a first integral of the given equation, 
we have, after an easy reduction, 
du Ou du du ou ou Ou 
5945 dr dx + = one (+ (5) 4 
dx or at or ° 
AT ou SE ou me: du 
år ore Då m 
ER. AR; 5 
Eliminating the ratio a between these equations, we find the fol- 
au |? du à pe Ou 
Ali] ne 
lowing equations: 
au 12 
Alt ou ou or | ‚| Me 
ds 3 ot ar Er ? - + (9) 
au ou Ga ou ou Ou 
a5) äh RE pep] 
Hence u, considered as a function of x, y, 2, ~, g, 7 $,4, satisfies the 
three given partial differential equations, which are all of the first order 
and second degree}. 
As # and v enter symmetrically into the equations (4), v will 
also satisfy three partial differential equations of the same form. The 
only condition respecting the application of the above equations is 
Å : å : à 3 
that we do not admit any relations, which make either zus or 
OG disappear. 
ot 
3. Proposition. The solution of the system of partial differential 
equations established in the last proposition, may be made to depend 
upon that of simultaneous linear partial differential equations of the 
first order. 
We observe at once, that if Do, the given partial differential 
equations are immediately resolvable into linear factors; we assume there- 
fore in the following, that D + o. 
Multiply the second equation of system (9) by an indeterminate 
quantity A, and add it to the first; we then have 
3 Ou ou du ou eu ou 
az] NE RA) at Sie oe er er or a JPA De (20) 
1 In the next section, we shall see that these equations are only a special case of a 
system of partial equations, determining a first integral of the general form of a partial 
differential equation of the third order, possessing a first integral, 
