PARTIAL DIFFERENTIAL EQUATIONS. 
191 
x dr + y ds + dp = 0, 
x ds + y dt + dq = 0, 
x dp + y dq -j- 2 dz = 0, 
to reduce to the same relation linear in x and y. That is, we must solve the system 
ds — \dr, dt = X 2 dr, dp — pdr, 
where A, p are given in terms of p>, q , z, r, s, t by the relations 
0F, , . 0F, ( . o 0t\ 
0r +X 0 S 
0F, 
0F 
, 0F, 
+ x3 m 1 + F a;, +2 ^ a - 1 — °> 
’ 0S 
0£ 
0F 
0* 
^2 
0F„ 
0F 2 0F S 9F 0F 0F 3 0F. 3 
+x a7 + x eT + ^ar + ^aT + if 1 sr = 0 ’ 
04 
0: 
04 
and q, z in terms ol p, r, s , £ by the relations F 1 = 0, F 2 =0. 
The complete primitive of these ordinary equations will involve three arbitrary 
constants, and there may be singular solutions with a lower number ; none of these 
will therefore constitute a complete primitive of the partial differential system 
F 1 = 0, F 2 = 0. 
