PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 253 



If a'', /3i, 7^ be the roots of the equation z^ + az^ + 0z + c=O, the solution is 



And a similar process applies to equations of aU 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 (rf"'rf") x (d"'d") u 

 may be written (rf™ x d"-f . u, in which form it is equivalent to rf-"' d-" .u: but the 

 combination (rf^a?") x (c?'"a^") . ?«, when written (as we shall write it) (d^'x^y . u, is not 

 equivalent to d-'"x-'' . u. The commutative law, or the law according to which 

 operations may be taken in any order, is not true of the symbols rf"", 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 or on X, giving the solution 



w= — J r; — I — —, for instance; the operation — ; , ■, ^ is resolved 



^ (Di-ai)(D*-;8i) ' ^ (Di-ai)(Di-/3i) ^^''"iveQ 



into the two operations ^^_ ^ ^IT^ ~ a^-Rl "d^W ^^ *^^ ^^^^^ manner as a 



fraction is resolved into its equivalent partial fractions. On this subject the 

 reader may consult an excellent paper by Mr Boole, in the Cambridge Mathe- 

 matical Journal, vol. ii., p. 114, where this method is first employed. 



10. Ex.4. ^+a^,+ 6y=X. 



dx dxi 



This gives y=(rfi-ai)-i(</i-/3i)-i . (X + 0) 



X 1 X 1 X 



Now 



(di_ai)(di-^) ai-(3idi-ai ai-jSi di-(3i 

 y = Ae"^ + Be''^ + -y^[(rfi-ai)-iX-(rf'-;8=)-ix| (Ex.3.) 

 Cor. 1. If X=:2Je": 



r ' 



2,= A e'^ +B /^-i- ^1-2 b /' C^i^ - -^-r) 

 ^ ai-^i •'■ \ri-ai »-*-/3M 



VOL. XVI. PART III. 3 S 



