24? Mr. Herapath on the Integration 



being a root of the equivalent algebraic equation, the part 



ace rx must vanish when the successive differentials of the 

 value are taken and respectively multiplied by A, B, C to an- 

 swer the conditions of the given differential. 



Therefore the part ae TX J Xe~ rx must also satisfy the dif- 

 ferential, that is 



Cae rx /Xe- rx = Cae rx fXe- rx 



Bd.ae rx fXe- rx = Bare rx fXe- rx + BaX 



Ad*.ae rx fXe- rx =Aar*e rx fXe- rx +AarX + AadX 



d?.ae rx /Xe- rx = at*e rx ' fXe~ rx + ar 2 X + ard X + ad 2 X 

 or (r 2 + Ar + B) aX + (r + A) adX + ad 2 X = X 



the other four terms vanishing in consequence of r being a 

 root. Putting for A its value = — r — r, — r ti and for B its 

 value =rr I + rr 2 + r t r 2 according to the theory of algebraic 

 equations, and restoring the value of a, the above equation 

 becomes 



r^X — (r, + r 2 )dX + d 2 X = {f—rr 2 — rr x -f r x r 2 ) X 

 or d*X - (r, + rJdX -(r 2 - rr.-rr,) X = 



a linear equation of the second order with respect to X. This 



equation therefore limits X to a particular form (namely e bx 

 b being determined from b 2 — (r, + r 2 ) b = r 2 — r?\ — rr x ) y 

 whereas it ought to remain indefinite. Consequently the so- 

 lution which^ requires this limitation cannot be a solution to 

 the equation proposed having X unlimited. 



In looking round for the cause of Lagrange's failure we 

 perceive that it must arise from the limitations which his equa- 

 tions of condition introduce. He assumes that the functions 

 p, p x , p 2 are arbitrary, and hence thinks the limitations he has 

 given them will not affect the accuracy of the solution. If, 



however, we consider that e rx is not a solution of the equation 

 proposed, but of this equation deprived of its term X, it is ob- 

 vious that the functional factor p has certain conditions to 

 fulfil, namely, to render the solution e rx of 



d 3 y + Ad z y + Bdy + C y = 

 a solution of the equation 



d 3 y + Ad*y + Bdy + Cy = X 

 and if at the same time we consider that the difference be- 

 tween the solutions of the former and latter equation depends 



entirely 



