1900. No. 5: ON PARTIAL DIFFERENTIAL EQUATIONS, ETC. 39 
Example. Let the given equation be 
—g.a+(1+p—g)B + (1 + Dy + (6 — ay) + (7? — 80) —(ad — fy) + p=0. 
Here Ar ER — (51.97), Ga DE OE AIG MIE: 
The equations for « and A become 
++ (1 + på +2=0 
f+ (1 + pu +2=0, 
which give 
h=— lg =— 5 4 =—A = — 15 
and when J 
AD — HG 
Au m = 
we must combine À, and wg or dy and w. 
Using A, and wg, the total differential equations for determining # become 
dr + dy + des=0 
dt + pdx + qdy + ds=0 
dp — rdx — sdy = 0 
dp — sdx —tdy = 0 
dz — pax — pay = 0, 
from which we find 
u=y+r+s=2,v=2t+5+t=35. 
A first integral of our equation is then 
y+r+s=f(2+35+%2) 
We shall now, in the manner here explained, proceed to deduce the 
systems of total differential equations corresponding to the various systems 
of linear partial equations established above. 
1. Letting Z=o, F + 0, G + 0, we had the sytsems 
ou Qu 
TO 
AF + BG ou eu ou 
Sen rat se |= 
where « was determined by the quadratic 
FGu? + CGu + HG —AD=o, 
