The Rev. Charles Graves read the following note " on 

 the Theory of Linear DifiFerential Equations." 



The equation 



2)"y + ^iD"-V + ^2-D"-2y + + Any^X, (1) 



in which yi„ ^2. • • • ^n, 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 = Z)-"X 



+ Co + CiX + . . . c„.,a;""* ; 



and this may be written as follows : 



(^{y) = D^X + Co + CiX + c.2X^ + . . . Cn-xX""'^ ; 

 if we employ to denote the complex distributive operation 



1 + D^AiD"^ + D-"A2D"-K . . + Z) "J„. 

 Operating now with the symbol 0"^ upon both sides of 

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

 posed one in the form 



y = ^-\D-^X) + Co<l>-\\) + Ci,p-\x) + Citp-'ix') . . . + c„_,^'-'(a;"-'). 



The term 0"\D""X) is evidently a particular integral of 

 the proposed equation ; whilst ^"' (I), <^'^{pc) . . . (^''(.r"'') are 

 particular integrals of the equation 



Z)"«/ + AiD"-^y + A.2D"-^y + ... + A„y = 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, Uo, Ml, M2 • • • Mn-i in place of (j)'\D'"X), ^''(1), 

 <j>'^{x), j>'\x^) . . . ^"'(a;""'), we may write 



y- U=CoUo + CiUi + . . . +c„.iM„.,; 

 and differentiating this equation n times successively we have 



