360 
NOTE ON THE PARTIAL DIFFERENTIAL 
[941 
viz. we have 
AD' - A'D = m,0 : 5(7' -B'C =- to,©, 
BD'-B'D = - m, CA' — C'A = - 50, 
(75' -CD = - 7®, .45' - A'5 = (7®, 
values which give 
(AD' - A'D) (BC - B'C) + (55' - B'D) (CA' - C'A) 
+ (CD' - C'D) (A 5' - A'B) = ® 2 (- mi m 2 +TR- VU) = 0, 
as it should be. 
Hence, when the partial differential equation ® = 0 is satisfied, we have 
d(u) d(v) d(u) d(v)_Q 
dx dy dy dx ’ 
and we thence have u=f(v) as the integral of the partial differential equation. 
It should be possible to express analytically the conditions in order that the 
systems of differential equations may have one or each of them two integrals. 
It is interesting to remark that, if each of the two systems of ordinary differential 
equations has only a single integral, these two integrals do not lead to the solution 
of the partial differential equation. Consider, for instance, the case 
5 = 0, S = x + y, 5 = 0, 17=0, V=p + q; 
the partial differential equation is here 
(x + y) s -(p + q)= 0, 
which has an integral 
z = (x + y) {</>' (x) + y\r' (y)) - 2 {cj)(x) + (y)}, 
where <£, yjr are arbitrary functions: the equation in to is m 2 —m(x + y) = 0, the roots 
of which are to = 0, and m — x + y. 
For = 0, to 2 = x + y, the system of differential equations becomes 
dy = 0, 
— (p + q) dx + (x + y) dq = 0, 
— pdx + dz =0, 
which has only the integral y = const.; and similarly for m 1 = x + y, to 2 = 0, the system 
becomes 
dx = 0, 
— (p + q)dy + (x + y) dp = 0, 
— qdy + dz =0, 
which has only the integral x = const. And these two integrals y = const, and x = const, 
do not in anywise lead to the integral of the partial differential equation.