PARTIAL DIFFERENTIAL EQUATIONS. 
1G3 
dx r ^ r dy - 1 - 1 dz cp 1 dr r dp 2 clx r dq 1 clx r dq 2 dx r 
o (r = 1,2), 
the letter P being used to denote differentiation with respect to x 1 or aq> on the sup¬ 
position that the other is constant, while 3 indicates strictly partial differentiation. 
Since dpjdx x — dpjdx z , dqjdx 1 — dqjdx 3 , we find by eliminating the deriva¬ 
tives of p u q x , p. 2 , q 2 , that 
J(«i,Pi, ft, ft) + Pi%,ft n ft, ft) + ft J (b Pi, ft, ft) + J (ft 2 ,Pa, ft, ft) 
+ Pa%, Pa, ft, ft) + ftJ( 2 ,Pa, ft, ft) = 0, 
and J(ajj, gq, p l5 pd) -f- p v J(y, p 1; p l5 p 2 ) + ftJ(?, ft, Pi, Pa) + J(%, ft, Pi, Pa) 
+ Pa%, ft, Pi, Pa) + ftJ( z ,ft,Pi,Ps) = 0 
where J( ) denotes the Jacobian of any four of the functions f l} f 2 , f 5 , f G with 
respect to the variables sjoecified in the bracket. Of these equations there are thirty, 
but since they are given by the elimination of six quantities from twelve equations 
only six of the thirty can be independent. 
§ 15. One pair of these auxiliary equations will contain Jacobians of f x , f 2 , f s , / 4 , 
and will in fact express the conditions that the equations 
dy = p l dx 1 + pydx 2 
dz — q l dx 1 + q 2 dx 3 
shall be integrable without restriction when p x , p. 2 , gq, q. : have the values given by 
the equations ^ = 0 =f 2 ,f s — a^f, = « 2 . 
Thus, if a pair of functions f 6 , can be found satisfying these two auxiliary 
equations, the solution can be completed by solving a pair of simultaneous ordinary 
equations. (See Mayer’s method, Forsyth, ‘ Theory of Differential Equations,’ pp. 
59-62.) 
The two auxiliary equations that f 3 , f\ must satisfy are linear and homogeneous in 
their Jacobians, the coefficients of the Jacobians not involving the functions f s , f x ; 
the number of independent variables is apparently eight, but it may be taken as six, 
since two of the eight variables aq, aq, y, 2 , p { , p z , gq, q. z are given as functions of the 
other six by the relations Jq — 0 ,f 2 — 0 , and may be supposed eliminated from 
if that is desirable. 
The columns of the matrix formed as at § 11 are the rows of the following array :— 
P(aq, aq), 0 , 0 , 
d(x x , y), 0 , 0 , 
d(x } , z), 0 , 0 , 
d(x 2 , y), 0 , 0 , 
(5) P(aq, z ), 0, 0, 
Y 2 
