1900. No. 5. ON PARTIAL DIFFERENTIAL EQUATIONS, ETC. 5 
form # = f(v), but also that the converse proposition is false. We propose 
therefore, first, to inquire under what conditions an equation of the second 
order of this form leads to a linear equation, and secondly, to establish 
upon the results of this direct inquiry the inverse method of solution. 
Proposition. A partial differential equation of the second order of 
the form u= fv), u and v being determinate functions of x, y, 2, p, I, 
r,s, t, and f an arbitrary functional symbol, can only lead to a partial 
differential equation of the third order of the form 
(3) Aa + BB + Cy + Då + H— 0, 
when u and v are so related as to satisfy identically the three conditions: 
D(u,v) ou ov du ov _ 
Dass er Wer eos: 
D(u, v) _ Ou dv Ou dv _ 
POI UE CC ETC NE 
D(u,v) _Ou dv: Ou dv _ 
DGD) or of Vet Ore 
Since w=/(v), we have du — f'(v) dv, an equation which, since /(v) 
is arbitrary, involves the two equations 44 — 0, dv =o. Hence 
2 5 Ot det, tapt ay + gg dry ds +S dt=o. 
zz DH det, dp +. cdg +e dr +o ds + a, 
But dz = pdx + gdy, dp = rdx + sdy, dg = sdx + tay, 
dr = adx + Bdy, ds = Bdx + ydy, dt = ydx + ddy. 
Hence, by substitution, 
ou 
ou Ou Ou Ou 
ee 
° à Ou ° 
EE ten tha tra tea 
a 
97 oz à 2 a ov à 
Er Er 
©) 97 oz 
testet te tea tra toa] v=o. 
Eliminating dx and dy, it will be found that the only terms involv- 
ing a, ß, y, Å in a degree higher than the first, will be those which 
