SATISFYING LINEAR DIFFERENTIAL EQUATIONS. 5 



matrix whose elements are series of positive integral powers of t, reducing for t = 

 to the matrix unity. 



4. Owing to the much greater simplicity of the case in which the equation 

 \a p+l p\ = has all its roots different, it will be treated first separately. The result 

 obtained is as follows : 



A matrix x can be determined uniquely of the form 



Xi+i i XP i i Xi 



tn+ 1 ' j i * * i , ) 



where XT>+I ---Xi are Matrices of constants in whicli all elements save those in (he 

 diagonal are zero, such that there is a formal solution 



where the matrix rj is made up of series of positive integral powers of f generally 

 diverging and reducing for / = to the matrix unity. 

 Consider the equation 



(B) $- - 



where 



r = I.,..., p+\. 



The roots of ot p+l p\ = being unequal, the matrix y)+1 will have zero elements 

 except in the diagonal; the diagonal elements will be /3 1; p.,, .../>, the roots of the 

 equation. 



If the equation (B) is satisfied by the matrix 



i) = (x, y, z...}, 

 where x, y, z ... denote columns of elements of the form 



X = X 



y = y+yit+..., 



the coefficients x r , y r ... being columns of constants x r a , x r \ x r 2 , &c., these constants 

 satisfy the following equations : 



X. (a p+l 6 1 p ^ l )x = 0, 



a +1 -0nz, + -<V)* = 0, 



