of Linear Differential Equations. 423 



nomial coefficients of the 2d, 3d, 4th, . . . coefficients of the 

 «th and « — 1th powers respectively, the equation becomes 



A2 



2 r^-'i- 

 n' 



n— 3 



A3 



3 ?-«-2 



,.n-3 

 'n—\ 



n — 4 





C3 



'n-y 



d}p + . 



d p 



or, X,= ^-ii5 + A,rf"-2p + B,rf«-3p + . . . Q,p (9) 



after integrating and representing c +fXe-''^ by Xj and the 

 coefficients of the above development by A, , B C 

 Now this being a linear equation with respect to j9 of one* de- 

 gree lower than the one proposed (7), may by a similar pro- 

 cess be depressed to another, 



X, = d-y, + A,d—3p^ + B,d!^-\ + . . . _p^^^ 

 wherein/*,, Xj are new functions of x, and A.,, B^, . . . other 

 numerical coefficients. In the same way this last may be de- 

 pressed to 



X3 = d'^-^ + A,d^-% + B.d'^-^p^ + . . . 0,p, 

 and so on continually to 



^, = d''-'p,_,+ A^d^-^-'p^_^ + B,d— Vi + --.(10) 

 and if ^ = w at length to 



and X,^ =Pn-i (11) 



one of the numeral coefficients disappearing at each step of 

 depi'ession. 



Again, we have seen that if r be a root of (8), the equiva- 

 lent algebraic equation of (7), 



X, = c+/X^-"^ 

 And by the same assumptions we have 



X, = c, +fX,e-'^ 

 X,= c, +/X,e-^^' 



X = c +/'X r'^t-a /|2) 



t i-\ J t-i ^ ' 



supposing R, Rp R^, . . . \ii_^ are respectively the roots of 



the 



