1894.] Ml' Brill, On Linear Differential Equations. 201 



(2) On the Application of the Theory of Matrices to the Dis- 

 cussion of Linear Differential Equations with Constant Coefficients. 

 By J. Brill, M.A., St John's College. 



1. Let p and q stand for the two matrices 

 "- ^^ and «- ^^ 

 Then we have the three identities: 



pq + qp- (a^ + K)P - («; + ^i) 9 



If we now form the identical equation satisfied by the matrix 

 X + py + qz, we shall obtain 



{x -[-py + qzY -[2x + y {a^ + S^) + ^ (a^ + K)\ i^ ^PV + ^^) 



+ (« + a^y + a^z){x + 8^2/ + V) " (/3i2/ + /^a^) (7i3/ + 7/) = 0, 



which may be written in the form 



[-x + {p-a,-h;)y + {q-a^-h^ z] {x+py + qz) 



+ «^^ + (aA - ^.7x) / + («A - /527.) ^^ 



+ {aX + a^ - /3,72 - /327i) y^ + («2 + ^2) ^^ + («i + ^1) ^y = 0- 



Thus we see that the theory of matrices will enable us to 

 resolve the result of the multiplication of the matrix unity by a 

 scalar ternary quadric into linear factors. In fact, we have 



ax^ + hy^ + cz"^ +fy^ + 9^^ + ^^^2/ 



= [ax + {h -ap)y + ig- aq) z] {x +py + qz), 



where p and q identically satisfy the three equations 



ap"^ — hp + b = 0, 



aq^ —gq + c = 0, 



a {pq + qp) —gp — hq +/= 0. 



It is further to be remarked that these two factors are com- 

 mutative. 



The above result enables us to replace the operator 

 «ar2 + ^o-2 + c^2+/a:7aI + 5'a7^ + ^ 



dx^ dy^ 9/ •' dydz ^ dzdx dxdy 



