4 ALF GULDBERG. M.-N. Kl. 
(1) 4a+ BB + Cy + Dd + Ei — ay) + Fy? — på) + 
+ Glad — By) + H= 0, 
in which A, B, C,.... G, A are functions of x, y, 2, p, 7, 7, S, t. 
The given partial differential equation of the third order is remark- 
able as including all the cases in which a partial differential equation of 
the third order admits of a first integral of the form 
(2) u=/A0), 
u and v being definite functions of x, y, 2, ~, 9g, % 5, & and /(v) 
arbitrary in form. 
We propose now to shew, first, that the solution of the given equa- 
tion on the assumption that a first integral of the form # =/v) exists, 
requires the satisfaction of a system of four! partial differential equations 
of the first order, of which one is of the first degree, the three others of 
the second degree; secondly, that this system, under given conditions, may 
be resolved into a certain number of systems of three partial differential 
equations of the first order and first degree, some of which are irrelevant, 
and others relevant, to the solution of the given equation; thirdly, that 
the solution of the relevant systems ultimately depends on the solution 
of a system of total differential equations of the first order and first degree. 
But before treating the general equation of the given form, we will 
deal, in a first section, with the linear partial differential equations of the 
third order. 
As to the methods employed in the present paper, they are essen- 
tially the same as those which Boole has used in his researches upon 
the partial differential equations of the second order of the Monge- 
Ampere form. 
The chief results of this paper have been summarized in a note pre- 
sented to the Academy of Sciences in Paris, on the 28th May, 1900. 
On the linear partial differential equation of the third order, 
Aa + BB + C + Då + H= 0, 
in which A, B, C, D, H are given functions of x, y, 2, p, 4, 7,5, &. 
1. It is easy to prove, not only that it is not true that a linear partial 
differential equation of the third order necessarily has a first integral of the 
1 If the given partial differential equation is linear, the system consists of only three 
partial differential equations of the first order and second degree. 
