ON THE THEORY OF PARTIAL DIFFERENTIAL EQUATIONS. 
136 
[656 
or introducing a well-known notation for functional determinants, and expanding the 
determinant, this is 
But in the same way 
d(a 0 , b 0 ) [3(H, c) 3(H, c) dq] ^ 
3 (p 0 , ?o) (9 (p, «) + 9 (p, q) dx] + ^ 
9(«n, b 0 ) J3(H, c) 0(H, c) dp\ o _ ft . 
9 (p 0 > q 0 ) (9 (q, yyd (q, p) cfyj + ’ 
adding these, attending to the value of (H, c), and observing that — C } = — ^ C } 
d(q, p) d(p, q) 
we have 
or 
9(a 0 , b 0 ) 
9 (p< 
the terms denoted by the &c. being the like terms with b, c, a and c, a, b in place 
of a, b, c. We have (H, a)=0, (H, b) = 0, (H, c) = 0, and the equation in fact is 
y d(a 0 , fr 0 ) 0(H, c)j fdq_^P) ==n . 
( d(p, q) 0(p, q)S \dx dy) ’ 
viz. we 
have = 0, the condition for the exact differential. 
dx dy 
Coming now to the case where the variables are x, y, z, p, q, r, and in the six 
equations treating p, q, r, p 0 , q 0 , r 0 as functions of the independent variables x, y, z,— 
then differentiating with regard to x and proceeding as before, we find for ~ the 
QjX 
equation 
9(c 0 , d 0 , e 0 ) (dr d (a, b, H) d(a, b, H)j + &c _ 0 
9(p 0 , % r 0 ) [dx d(r, p, q) d(x, p, q) 
We have, in the same way, for ^ the equation 
0(c o , d 0 , e 0 ) (dp d(a, b, H) 0(a, b, H) 
9 (p 0 , q 0 , n) idz 0 (p, r, q) + d(z, r, q) } + &C ‘ °’ 
or, adding the two equations, 
0(c o , d 0 , e 0 ) \(dr__ dp\ d(a, b, H) d(a, b, H) d(a, b, H)| 
9(po, ?0, n) l\d® ife/ 0 (?’, p, <7) 0(fl7, p, q) d(z, r, q) j + C '~ ’ 
where the terms denoted by the &c. indicate the like terms corresponding to the 
different partitions of the letters a, b, c, d, e. 
The equation may be simplified ; we have identically 
-P<k H)-®(H, 
dq dq dq 0 (x, p, q) 0 (z, r, q)