Differential Equations of the Third and Higher Orders. 69 



where F(xy) = c is the integral of = RacZz/ — Tdx ; and 



. Ya 



Fi(?/c) is the result of elimination of x from the quantity -7^- 



by means of ¥(xy) = c. 



The adoption, in the case of equations of the second order, 

 of the method of this paper not merely gives us the first inte- 

 grals resulting by Mongers method, where such exist, but 

 develops explicitly the conditions of their existence. It does 

 more : for to the carrying out of Mongers method it is essen- 

 tial that the coefficients R, S, T, V should either be functions 

 of x and y only, or of p and q only ; whereas the method 

 above set forth equally gives the first integrals where they 

 exist, and the conditions of their existence, when R, S, &c. 

 are functions of x, y, p, q ; the only advantage of the restric- 

 tion of the coefficients to being functions of x and y only, 

 being that it enables us to carry further the development of 

 the results when the coefficients R, S, &c. are expressed by 

 general symbols. 



The same remarks hold when the method is applied to equa- 

 tions of a higher order than the second ; when R, S, &c. may 

 involve, in addition to x and y, any differential coefficients of 

 an order lower than that of the equation to be integrated ; 

 although when R, S, &c. are so made up, the cases in which 

 first integrals exist will probably be very rare. 



6 New Square, Lincoln's Inn, 

 July 20, 1885. 



P.S. — The law of formation of the equations in a will be 

 seen by considering the case of equation (9), in which R and 

 W are the initial and terminal coefficients of the left side of 

 the equation. Here the 1st, 2nd, 3rd, and 4th coefficients of 

 the equation in a will be multiplied by R 4 , R 3 , R 2 , R ; and 

 the 3rd, 4th, 5th, ^and 6th by W. W 2 , W 3 , W 4 respectively. 

 Each of the terms from the 2nd to the 5th, both inclusive, 

 will involve one other factor, viz. V, U, T, S respectively, i. e. 

 the coefficients of (9) intermediate to R and W taken in 

 reverse order. 



The late Mr. Boole showed that the fact of the general 

 equation of the second order admitting of a first integral of 

 the form u = cf>^v), where u and v are definite functions, and 

 $ is arbitrary, is conditional : but his method fails to develop 

 the conditions. 



