On Partial Differential Equations of the 
Third Order 
by 
Alf Guldberg. 
The theory of partial differential equations of the first order in one 
dependent and any number of independent variables may be regarded as 
complete in so far as we regard their theory as fulfilled, when we are 
able to reduce their integration on that of a system of ordinary differen- 
tial equations. 
The same has been secured by the classic works of Monge and Ampere, 
for the theory of partial differential equations of the, second order in one 
dependent variable and two independent variables, when the given 
equation admits of an intermediary integral, — a result which has recently 
been generalized by Vivanti and Forsyth to partial differential equations of 
the second order in one dependent variable and three independent variables 
possessing an intermediary integral. 
The present paper deals with those partial differential equations of 
the third order!, involving one dependent variable (say z) and two inde- 
pendent variables (say x, y), and possessing an intermediary integral of 
the second order. When the derivatives of z of the first order with regard 
to x, y are as usual represented by g and g, those of the second 
order by 7, s, ¢, and those of the third order are taken to be «, B, 7 0, 
the general form of the equation, the theory of whose solution we 
propose to consider, is 
1 The results may, without difficulty, be generalized to the partial differential equations 
of the th order, that admit of an intermediary integral. 
Vid.-Selsk. Skrifter. M.-N. Kl. 1900. No. 5 
