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XVI. On the Error imputed by Mr. Herapath to Lagrange 

 in his Method of integrating Linear Differential Equations. 

 By A Correspondent* 



To the Editors of the Philosophical Magazine and Annals. 

 Gentlemen, 

 XTOUR Correspondent Mr. Herapath has endeavoured to 

 A prove that Lagrange has fallen into an error in his in*- 

 tegration of linear differential equations, and expresses his sur- 

 prise that mathematicians have for upwards of half a century 

 suffered this to pass unobserved. Skilful mathematicians will 

 not take Mr. Herapath' s word for his bold assertion without 

 examination, and will then easily detect the fallacy of his rea- 

 soning : but as your readers are not all of them mathemati- 

 cians of competent knowledge, it will perhaps not be useless 

 to point out Mr. Herapath* s error. 



Mr. H. has deduced the following equation (page 24?.) 



{f + Ar + B) aX + {r + A) adX + ad 2 X = X 



a being : = ; -77 ; 



This equation is quite incomplete. Mr. H. having employed 

 that part of y which depends on the root r, only; all the terms 

 arising from those parts of y which depend on r, and r 2 are 

 left out. A and B involve the three roots equally, and their 

 values may be substituted in the above terms on the left-hand 

 side of the equation as Mr. H. does, and we obtain for them 

 the following 



r^aX — (r t + r 2 )adX -f ad 2 X 



Call a\ a" the values corresponding to a for the two other 



roots r., r q , so that a 1 == .— r and a"= «*% 



(r, -r) (r x -r 2 ) (r 2 -r) (r 2 -r t ) 



The terms which are, consequently, to be added to make 

 Mr. H.'s equation complete will be 



r r 2 a'X - (r + r 2 ) a! dX +a'd 2 X 



rr x a"X — (r+ r x )a"dX + a" d l X, and the aggregate 

 of all terms will give this equation : 



(r.r^+rr^' + rr.a'OX-Un + rJa + ^ + ^y + Cr-frXJ^X 

 + ( fl + a'-f a»)^X^X. 

 It will now be easily seen that a + a! -f #"= 0, 

 (r t + rjfl + (r + r 2 ) a' + (r + r x ) a"= and 

 r x r 2 a + r r. 2 a' + r r, a" = 1 and that, consequently, the equa- 

 tion becomes X = X, as it ought to be. This is the same 



New Series. Vol. 3. No. 14. Feb. 1828. O proof 



