1900. No. 5. ON PARTIAL DIFFERENTIAL EQUATIONS, ETC. 
7 
Hence, if we represent each member of those equations respectively 
with #, » and /, we have 
Ou Pr © Ou ay Ou 
Os dr’ ot or” 
Å (0 
v ov ov ov 
== 00 le if 3 
Substituting these values in (4), we have 
sets dy +o dz = +; dat, “(dr + mas + ldt) = 0 
de and 4 de + dø + dg +s (dr + mds + Id) — 
Now making dz = pdx ni gay, dp =rdz + sdy, dq = sdx + tdy, 
we have: 
tems ea 
+ 2% dr 4 mds + Id) = 
(6) 
ov ov 
Ge pate el wur vn: 
432 (dr + mas + Id) = 
From these, and from the equations 
dr = adx +Bdx, ds = Bdx + ydy, dt = ydı + day, (7) 
if we eliminate the differentials dx, dy, dr, ds, dt, we shall necessarily 
obtain a result of the form (3). 
To effect this elimination, we have, from (7), 
dr + mds + ldt = (a + mß + ly) dx + (8 + my + 18) dy, 
or 
adx + 8 (dy + mdx) + y (dx + may) + dldy — (dr + mds + /dt — 0. (8) 
Now the system (6) enables us to determine the ratios of dy and 
dr + mds + ldt to dx, and these ratios, substituted in (8), reduce it to the 
form (3). 
But in order that it may be not only of the form (3), but actually 
equivalent to (3), it is necessary and sufficient that we have 
dx dyt+mdx _ldx4+ mdy ldy _dr+mds+ dd 
79 TN ES 
