PARTIAL DIFFERENTIAL EQUATIONS. 
1.83 
(15) cl(x, t), — R 0 
d(y, r), 0, - T 
d(y, s), T, 0 
d(y, t), - S, R 
d(x,p), 0, 0 
(20) d(y,p), 0, 0 
d(z,p), 0, 0 
d(x, q), 0, 0 
d(y, q), 0, 0 
d{z, q), 0, 0 
(25) d(p,q), 0, 0 
d(x, z), 0, 0 
d(y,z), 0, 0 
d(x, y ), 0, 0 
X, P . . . are written for dj/dx, df/dp . . . 
Of these twenty-eight rows, only twenty-one are independent. For instance, 
multiply the 1st, 2nd, 4th, 7th, 10th, 13th, 16th by — S, — T, P, Q, Z, X, Y respec¬ 
tively and add ; the resulting row is 
d(f r), 0, 0, 
which vanishes since f = 0 by hypothesis. 
Suppose d (m 15 u 2 ) to be the complete bidifferential formed from the determinants 
of the array, then to complete the solution we have to find r , s , t from the 
equations 
f — 0 , u x —■ 
and integrate the equations 
dz — pdx + qdy> dp — rdx + sdy, dq = sdx -j- tdy c 
It will amount to the same thing if we treat u x as known in the auxiliary 
equations. They must be satisfied if u z , iq, u 5 are substituted in turn for w 3 . Now 
two homogeneous linear partial differential equations in seven independent variables 
can at most have five common solutions, and here one of these, ?q, is known ; the 
other four may be taken as u 2 , u 2 , u±, u- 0 . 
§ 33. Any two of the five functions x, y, z, p, q will satisfy the auxiliary equations, 
but as we have to solve for r, s, t, these solutions will not serve our purpose. They 
are ten in number, and ten more will be given by taking in pairs the expressions u x , 
Mo, m 3 , u 4 , u 5 given by any complete primitive. These twenty are not all bifunc- 
tionally independent, for since there are three relations* among the ten expressions 
x, y, p, q, «i, u. 2 , m 3 , m 4 , u-, 
* Compare § 34, p. 184. 
