
PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 2583 
If a=, 63, y2 be the roots of the equation z+ 2?+62+¢=0, the solution is 
y=A e** + Beh* + Cer 
And a similar process applies to equations of all orders, with constant coefficients. 
9. It will be seen that in solving these equations, we treat symbols of ope- 
ration in exactly the same way as if they were symbols of quantity. Our jus- 
tification for so doing is an appeal to the fact, that the laws which regulate 
the combination of the former symbols are precisely the same as those which 
regulate the combination of the latter. Were it otherwise,—were one of the sym- 
bols, for instance, to be subject to a different law relative to its combination with 
one class of symbols from that which regulates its combination with another, we 
should not be at liberty to separate the operations of such symbols, nor even to 
combine them otherwise than in the form in which they are actually presented 
to us. An example will illustrate this remark. The combination (d”d”) x (dd”) . u 
may be written (d” x d”)?. vu, in which form it is equivalent to a?” a?".u: but the 
combination (d”x”) x (d”x").u, when written (as we shall write it) (d™x”)? . u, is not 
equivalent to @"2".u. The commutative law, or the law according to which 
operations may be taken in any order, is not true of the symbols a”, x”, in their 
combination with one another. 
We may remark, in addition, that when an operation on y has been changed 
into the reciprocal operation on 0 or on X, giving the solution 

if 
1 : - il : 
= = —h6?7>) for instance; the operation : 
(D!_ a) (DIB) 0, p (Dia) MB) is resolved 
: . 1 1 iE ik ; 
into the two operations apt Di oak af g) DE er in the same manner as a 
fraction is resolved into its equivalent partial fractions. On this subject the 
reader may consult an excellent paper by Mr Boots, in the Cambridge Mathe- 
matical Journal, vol. ii., p. 114, where this method is first employed. 

dy, #y hs 
10. Ex. 4. ae by =X. 
This gives y= (dt—a*)—1(d+— B2)-) . (X +0) 
N x eps ase nha?. Sit ar o's Saeenle Sae 
ey @—a) (@—B}) at_Bi@—at  ai—B! a—B? 
1 
yo=ANe** + Beet 

ema { (@t—a)-1X-(@-6) 1X} (Ex. 3, 
Cor. -1.. If X=abe; 
y=Ae* +Beo*4 

dt 1 1 
ae (a 
apt” \A-at 7p} 
VOL. XVI. PART III. OAs 
