The Rev. Charles Graves read the following note " on 

 the Theory of Linear Differential Equations." 

 The equation 



Z)"7/ + ^iD"-V + A^D^-^y + + /i„r/ = X, (1) 



in which Ay., A^, • . • An, and X, are any functions of x, and 

 D stands for the symbol — , may be brought, after n integra- 

 tions, into the form 

 y + D'^AiD'^-^y + D-^A^D^^-^y + . . . + D^Any = D^X 



+ Cq + CxX+ . . . Cn-.\X^'^ ; 



and this may be written as follows : 



<^{y) = D^X + Co + Cyx + CaOJ^ + . . . c„_ia;"-^ ; 

 if we employ to denote the complex distributive operation 

 1 + Z)^y4iD™-i + JD-MgD"-' . . . + D-^An. 

 Operating now with the symbol ^"^ upon both sides of 

 the last equation, we obtain the complete integral of the pro- 

 posed one in the form 



y = <t>-\D^X) + Co<i>-\\) + c,<ir\x) + c^<i>'\x-') . . . + c^,^»-Xa;»-'). 



The term (j)-\D^X) is evidently a particular integral of 

 the proposed equation ; whilst 0"' (1), (p'^^x) . . . ^-'(a?""') are 

 particular integrals of the equation 



D^y + ^iD«-V + A^D^-^y + ... + Any = 0. (2) 



This demonstration of the presence of n arbitrary con- 

 stants in the complete integral, and of the mode of its com- 

 position, seems more simple and direct than those which are 

 commonly given. 



Putting U, Mo, Ml, Ma . . . Un-i in place of ^-\D-"X), (j)-\l), 

 <l>-^(x), (p-\x^) . . . 0-'(a;"-'), we may write 



y- U=CoUo + CiU, + ...+Cn.iUn-i', 



and differentiating this equation n times successively we have 



