1900. No. 5. ON PARTIAL DIFFERENTIAL EQUATIONS, ETC. 13 
Eliminating # and ” from these equations we have 
å åg Adr + (D — 1,454) ds + 1,4, Hat —1,Ddt =o 
dy + 1,dx = 0 
dp — rdz — sdy = 0 
dg — sdx — tdy = 0 
dz — pax — qdy = 0, 
or written in another form, 
å, Adr + (43 A + å, B) ds — Dat + 1, Hdıx = 0 
dy + 1,d%=0 
db —rdx — sdy=o ( (13) 
dg — sdx — tdy =0 
dz— pgx — qdy =0 
This then is the system of total differential equations deduced from 
(12°) upon the integration of which the determination of will depend. 
If we here successively subititute A, and A, for A,, where A, and A, 
designate the two other roots of the cubic (A), we get the different systems 
of total differential equations corresponding to the systems of linear partial 
differential equations obtained in the same manner by (12). 
If one of these systems thus obtained admits two integrable com- 
binations, 
van =, 
it is obvious, from what precedes, that 
u =f (v) 
will constitute a first integral of the proposed equation (3). 
Remark We should also, by a more direct process, have obtained the 
three different systems of the form (13). For eliminating # and / between 
the equations (A), page 7, we find that 
Ady? — Bdy?dx + Cdydx — Ddx* =o 
Adxdydr + Bdxdyds — Ady?ds + Ddx?dt + Hdx?dy = 0, 
