of Lineal- Differential Equations. 425 



root of the equivalent algebraic equation (8). And because 

 this must be the case for every value of/ from to n — 2^ 



r ■=. r R = r, — r 



r'i=. r + R R, = ?-^— r, 



r^ = r + R + R, Rj = '•a — 'Ti 



?-3= r + R,+ R2 which give R3 = ?%— 7-3..(16) 



71 — 1 



ax y ■=» e x 



(IV) 

 which is the integral in a calculable form, free from all inter- 

 mediate reductions, and containing all the n roots of the 

 equivalent algebraic equation, and n arbitrary constants. 



By partially performing the integrations (17) is transformed 

 into 



xr « XT' - 



xrn-\ ^ n-1 1-3 ir 



71—2 71-1 71—3 n— 2 



"ti-I /'''('■ -»• ) /•i('- -r ) _,r 



+ ^ y^ n-27i-iye «-3 n-2 /^ X...(18) 



which in many cases is a more convenient form than (17), and, 

 as we shall hereafter see, particularly serviceable in establish- 

 ing the completeness of the solution. 



O:^ In consequence of the difficulty of printing correctly (17), (18), and 

 the f)rece(lint; foriniilii, the author informs tlie reader that each n, n— 1, 

 n— 2, . . . heh)n<;s to tlie ;■ ahove it, that i is a factor to each r, and the 

 whole product an ex[)onent to e; also in (18) that '^,,_i. C,_2. c _>^, are 

 each factors to its contiguous power of c. 



[To be continued.] 



New Series. Vol. 2. No. 12. Dec. 1827- 3 I LXXI. On 



