Mr. Herschel on various points of Analysis. 4^7 
and from these, ”a may be derived by considering that the 
comparisuii of the equations (1) and (3) gives 
I 2 W— I 
CC • CC m ^ 
, oj = A, or u = A . ^ a . . . a) 
Having obtained V, .... ^ccy nothing more is requisite for 
obtaining a complete integral of (i), than to substitute their 
values in equation (4). 
The method here delivered of obtaining the known theorems 
respecting the equation 
0 = + * ADw + . . . ”A . + X 
appears to have the advantage in point 
of conciseness over any I have hitherto met with ; a sufficient 
apology for the revival of a subject whose theory, and whose 
difficulties have been so long and completely understood. 
In the case when X = 0 and *A, . . . ”A are constant, the 
method of separating the symbols of operation from those of 
quantity, may be introduced with great elegance. 
Let/), q, r, &c be the roots of 
D 
«— I 
+ • • • + 
I 
”A 
and the equation ( 1 ) becomes 
0 = (D — />)(D — q) ... die: u 
which is satisfied by either of the equations' 
0 = (D — /)) : u, 0= {D — q) : u, &c. or, 
D// = pu, Du =■ qUf &c. 
Now these equations integrated give the following 
u U = f?', U = &c. 
which are the particular integrals of the proposed, and of 
course its complete integral will be 
«= 'C. + ^C. &c. 
