(5') 



Integration of Differential Equation of the Second Order. 1 19 



where f x ,f respectively denote the complete derivatives of/ with 



respect to x and y, taken on the assumption that p is constant. 



The two last equations embody all the relations between the 



d 2 z d^z d^z 

 partial differential coefficients -j-^t -p? i i ? ana * the variables 



30, y, z, p which are capable of being derived from (2). 

 But if we put/ for q in (1), we shall have in the equation 



°=V,-g + S g=/ .i| + T ?=/ .| + W (5) 



a relation between the same partial differential coefficieots and 

 variables which by hypothesis is derivable from (2). Hence the 

 right side of (5) must be identical with 



where A, B are functions of x } y, z, p ; in order to which we 

 must have 



B-A ./'(*>) =S S=/ , 

 -B .f(p) = -R q=f , 



whence, eliminating A and B, we get 



0=T ?=/ ./ 7 (^ 1 2 +S g= ,./'(rf + R ?=/j . . (6) 

 0=n q=/ .f-(T q=/ .f^ + Y q=/ )f'(p). ... (7) 



Let m„ m 2 be the values of f'(p) given by (6) ; and putting 

 m x foYf'(p) and dividing by T g= /, (1) becomes 



or 



0=M,/'(*)-/'(y) + K|»-/) ./'(*) -Q . . (8) 



The function / must satisfy simultaneously (8) and also the 

 equation 



0=f'(p)-m l (9) 



As an illustration of the mode of procedure necessary to find 



f } take the case where -^ is of the form up-\-fiq-\-<yz + 8, and 



all the coefficients R, S, T and a, /3, 7, 8 contain x and y only. 

 Integrating (9), we have 



f(xyzp)=m l p+f a (ocyz). 



