1894.] Discussion of Linear Differential Equations. 203 



which is equivalent to 



\ Aj —\ J 



2m -X —\ 



or — r H- =■ const. 



\-\ 



Now ii f(x) be any scalar function which may be expanded 

 in powers of ,v, we may, by substituting m for x, obtain a matrical 

 function which may be spoken of as framed on the model of the 

 scalar function f{x). Writing f(m) for this matrical function, 

 we have by Sylvester's Interpolation Theorem 



Differentiating this, we obtain 



If we assume for dm the value given by equation (1), then it 

 is easily proved that the earlier part of the expression for d ./"(m) 

 vanishes, and the above equation reduces to 



^ -/w = '^^/ (\) d\ + ^;/'(^) <^\' 



But, multiplying equation (1) by m — X^, we have, in virtue of 

 the identical equation satisfied by m, 



don . (m - X^) = ^ ~- c?\ 



Aj - A,2 



or (c^m — dXj) (m — \) = 0. 



Similarly we shall obtain 



{dm — dX^) {m — \) = 0. 



