PARTIAL DIFFERENTIAL EQUATIONS. 
175 
so that p x , q x are two functions satisfying the auxiliary equations, and the integral is 
to be found by putting p x = a, q x = b. Thus we have 
y — ctx l = F (x. 2 , a, b , p 2 , q 2 ), 
z — bx x — G ( x. 2 , a, b, p. 2 , q 2 ), 
V = F(£ a, b , V, 0, 
£ = g- (£r>, 
g = x. 2 , t/ = y — ax x , £ = z — bx x , 
V = dn/dt, £' = dt/d£ 
These are ordinary differential equations, the solution of which will involve two 
new arbitrary constants and so constitute a complete primitive of the original 
equations. 
§ 27. III. The equations 
or 
where 
y = p x x x + lh x -2 + <KPi>Pz> ?i> 
2 = q lXl + qyx 2 + \f/(p lf p. 2 , q x , q 2 ), 
are of special interest, because more complete primitives than one can be found. The 
obvious solution is p x = cq, p 2 = a 2 , q x = b x , q 2 = b 2 , 
y = cqaq + a 2 x 2 + a 2 , b x , b 2 ), 
z = b x x x + b 2 x 2 + a 2 , b x , b 2 ). 
Suppose a x , a 2 , b x , b 2 to he variable, but functions of one variable only— say ci x , then 
their variations must satisfy the relations 
x x da x —h x 2 clct.-t —(~ d(f> — 0, 
x x db x + x 2 db 2 + dxft — 0. 
These define x x , x 2 , and, therefore, also y, z as functions of a x , unless the determi¬ 
nants of the matrix 
da x , da 2 , d<f> 
db x , db 2 , d\p 
vanish ; it is necessary, then, that these determinants should vanish. Thus 
cq, b x , a 2 , b 2 are connected by two ordinary differential equations. We may assume 
any third relation connecting them at will; suppose b x = F(cq), F denoting an 
arbitrary function. Then by integration we may suppose cq, b 2 found in terms of cq. 
Also cq is connected with x x , x 2 by the relation 
cla 0 , (Uh 
Xi +*r; + = °- 
so that cq, b x , cq, b 2 are all known in terms of aq, x 2 , and by substitution the values 
of ?/, 2 are found. 
