44 PROFESSOR A. R. FORSYTE ON THE INTEGRATION OF 



where I is a positive integer, can be expressed in finite terms. To express the 



integral, let 



F (, 0) = F, where a = x + y, /? = z, 



G (', /3') = G, where ' = .r + , P = y \ 



~\ ? P 



and denote by A the operation g~ ~~ 90 > b y A ' tne operation ^, ~^> , so that 



8G G A ' T <- 8 - G 9 

 ~" -'*" 



Then the value of r is 



v = F + G 



+ A (y/ + z) (AF + A'G) 



+ , (y -r- :)' (A'F + A"G). 



24. But it is necessary to take account of Avhat has been achieved when one 

 equation or when two equations (say of the second order) have been obtained 

 compatible with the given equation and involving each one arbitrary function. 

 The method adopted in 11 to pass to the primitive has manifestly no element of 

 generality. 



Now the three equations are not sufficient to express a, 7>, c, f, g, h, in terms 

 of I, m, n, and the variables ; but they frequently will serve to express groups 

 of combinations of a, b, c, /, g, It, in terms of those quantities. Thus the three 

 equations in 11 suggest combinations a h, h b, g y (which are the derivatives 

 of I m), and a g, h f, g c (which are the derivatives of I n). This, 

 however, is only a slight modification of the former method ; it, again, has no element 

 of generality. 



Another plan would be to differentiate the three equation^ up to any order with 

 the hope of determining all the derivatives of the highest order that occur in 

 terms of derivatives of lower order. If this were possible, substitution in the 

 equations of differential elements such as 



dl = adx + hdy + gdz 



and successive integration would ultimately lead to v. It appears in general, how- 

 ever, that relations of interdependence among the equations prevents them from 



