57 



erroneous conclusion. But the form (6) is not sufficiently ge- 

 neral to include the case of /3 = o. For its general form I find 

 to be by the method above referred to, 



^-^ c, cos/3.r +('^1 + g3)sin/3a: + L^ ^ Bm^x-^^xco^fix\{7) 



This, because of the arbitrary quantities, agrees M'ith the form 

 (6) in every case but when /3 = o. 

 By making fi = o in form (7) the result is 



e"^ \c^ -{-c^x^c^ ar' +c^x^ X because when /3 = o 



sin S X sin/Sx x cos g x x 



-J- = X, ~p ^, — 1T2 



The next example is from Lagrange ;* it arises from the 

 integration of an equation of finite differences, and which, as that 

 author seems to thmk, furnishes a strong objection against deducing 

 the properties of differential equations from the limits of finite dif- 

 ferences. 



"' y ' y °^ • < be corresponding values 



X, X + I, X + 2 I &c. 3 ID 



of which the relations are 



y := a X -\. a' 



'!/ = a'{_x+ i) + a' (8) 



&c. &c. 

 and a' (x + + a' r: a (jt + t) + a^ (9) 



Then y' = a{x + i) + a^ 

 and Ay = y' — y = at 

 Hence y = ='4^ + ^^' O^) 



VOL. XHI. K 



* Seances des ecoles nofmales, 1801, p. lOl, &c.. 



