212 Mr. Herapath on the Failure of Lagrange's Method, Sfc. 



Lest it be supposed that the failure of Lagrange's method 

 may be remedied by putting the pretended " complete inte- 

 gral " under the form of our (17), so as to take away the zero 

 divisors, and thus bring out the complete solution : I shall now 

 examine this point. For the sake of simplicity I will take an 

 equation of the second order only, the lowest to which La- 

 grange's method can be fairly applied ; and which from its 

 having but two roots, and therefore but one difference between 

 them, is peculiarly adapted to exhibit perspicuously the justice 

 or injustice of our observations. 



If r, r x be the roots of the equivalent algebraic equation, the 

 " complete integral " is 



rx r.x . rx f^r —rx . r, x /*■«■ — r, x , 



_ c e —c,e ' -f g ./ Xe dx—e ' J Xe l dx 



Jr r —r l 



And if we put X = 



ce rx - c ,e riX r t x f (r-r x )x , rx / % ( r i- r > J 



y = —ce ' J e K dx -f c t e J e K ' dx 



AT r -r x 



must be the " complete integral " of 



dx* dx ^ 



But when r = r, the latter expression for the " complete in- 

 tegral " is 



y = (c + c^xe** 

 which is evidently not the " complete integral," were it only 

 that c + c x makes but one arbitrary constant. The integral 

 by our formula (17) is 



y = {c x + cx] e rx 

 in which two arbitrary constants appear. 



J. Herapath. 



P.S. Since writing the above, I have examined with more 

 attention Lagrange's method, and I find that the solution (b\ 

 which is professed to be taken from Lacroix's great work, as 

 the " complete integral " by Lagrange's method, is not strictly 

 the integral which this method gives. The result of La- 

 grange's contains less of absurdity than the above, but never- 

 theless fails to give the complete solution. I have not found 

 where Lagrange published his method, nor what examples of 

 it he gave. If that of Lacroix be one, it is a curious circum- 

 stance that he should fail both in method and deduction. 



XXXV. On 



