PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 



301 



= A (o2 + 1)' + ~ ~ X 



A — a-^ 



= A (a2 + 1)^ + (A* + a) (o^ + !)■' 2 



'(«2+l).^+i 

 Cor. 1. Let X=Aar; then the solution of the equation Aiuj. — au,=bx is 



Cor. 2. Let X=b 



a' z + a' + 1 

 x{x + l) 



then the value of (a - o^) ~ ^ X is — 



therefore the solution of the equation a*m, - a« =a° ^"'"" "'"^ 



IS 



M,= A(a2 + ir- (A4+a)- 



=A (a2 + i)-_^_6 vni^ii ^by Ex. 6, Art. 24.) 



Ex.7. AM^ + aA^M^ + 6M^ = 0. 



This equation may he written 



(A + a aJ + 6) M^ = 0. 

 Let a, /3 be the roots of the equation z^ + az + 6=0, then 



(Ai-a)(Ai-/3)«,=0 

 M,„= A(A*-a)-i . + B (a*-i8)-i . 

 = A (1 + a^)^ + B (1 + j3^y (Ex. 4.) 



Cob. If a=/3, we obtain, by the usual process, m^=(A + B.i:) (l + a'^K 

 Ex.8. Au^ + aAiu^ + bu^ = c(l + e^y' 



The solution is 



tt.=^i^(A*-a)-i(0 + ciTi2|-)--i^(A*-/3)-^(0 +cnV|-') 



= A(l + a2f + B(l + ^2j' + 



(a-/3) (e-a) (a-/3) (e-;8) 



= A (1 + a'-y + B (1 + /S^)-^ + <:0-+e^)l 



e^ + ae + b 



Cor. If e=a, we must proceed as in Cor. 3, Ex. 4, Class 1, of Differential 

 Equations, and we shall obtain 



,= A (1 + ar + B(i+^r+ '"^I'^^Y 



VOL. XVI. PART III. 



4g 



