of Linear Differential Equations. 25 



entirely on the influence of the quantity X, it is clear that p 

 which transforms one solution into the other must also de- 

 pend on X, and for this reason cannot be arbitrary. Con- 

 sequently the very first assumption of Lagrange is erroneous, 

 and the arguments which he founds on this assumption pro- 

 ductive of error. It is most extraordinary that mathemati- 

 cians have for upwards, I believe, of half a century suffered this 

 to pass unobserved and uncorrected ; and, as if to crown their 

 conduct with absurdity, have actually praised Lagrange's me- 

 thod as the most perfect and general that could be given. 



Verification of our own Solution. 



The simple manner in which we have, step by step, deduced 

 our general formula of solution cannot, we presume, leave a 

 doubt as to its truth. But since we have shown the failure of 

 Lagrange's method by differentiating one of his examples, it 

 is but fair that we should apply the same test to a similar ex- 

 ample given by our own method. 



It is obvious by (12) and (16) that 



— xR x(r — r ) 



dX t = X t _ ie <~2 =X f _ x e n-2 n-1 



and in the equation just above (17), that 



xr 



Therefore putting n = 3 we have 



dy=r 2 e r * x X 3 + e r > x X 2 



d*y=r 2 e r *X, -f (r 2 + r^e^X, + e rx X 



^=;Y^X 3 + (r a H^^ 



If we multiply these values ofd 2 y, dy, y by A(= — r 2 — r, — r), 

 B ( = i\r t + r. z r + r, r), C (= r 2 r, r) respectively, it is clear, 

 r 2 being a root of the equivalent algebraic equation, that 

 the sum of the left-hand column will = 0. It is also obvious 

 from the value of A, that the two terms which constitute the 

 third column = 0. But the numeral part of the second co- 

 lumn, putting for A and B their values, is plainly 



r* + r i r l + r* — {r 2 + r t ) . (r 2 + r x + r) + r 2 ?\ + r z r 4- r,r=0 



So that all the terms vanish except X, and leave X = X as it 

 should be. 



We might here develop some other properties of our so- 

 lution, both as it regards integration and even the doctrine of 

 algebraic equations; and we might show how to extend a 

 similar method to linear equations having variable coefficients ; 

 but these things we think it better to reserve for another op- 



New Series. Vol. 3. No* 13. Jan. 1828. E por- 



