346 PROCEEDINGS OF THE AMERICAN ACADEMY. 



The determinant of these n solutions can be written out in such a way 

 that the elements above the principal diagonal are all zero. Thus the 

 value of the determinant is : 



X , 



and the n solutions are therefore linearly independent. 



On account of the relation (10), each solution (11) of the system (8) 

 will determine a solution of the original system (3). Accordingly we 

 have n solutions of (3) which are linearly independent, for their de- 

 terminant at any point is equal to the determinant of the A' a in (10) 

 multiplied into the determinant of the solutions (11) for the same point, 

 and neither of these determinants is zero. 



Suppose now we consider any multiple root of the characteristic de- 

 terminant (5) ; for simplicity let us take r u and suppose that 



= r. 



to 



(12) r, = r 2 

 so that 



(13) «! + e 2 + . . . + e k 



is the multiplicity of the root r x . Then from (10) and (11) we see that, 

 corresponding to this root, there are k solutions of (3) not involving log x 

 and linearly independent : 



= 1,2,... » 

 1,2, . . .& 



04) ^"-=4,,^ (: 



Therefore the constants A- „ are linearly independent solutions of the 

 equations (4) when r = i\ , as we readily see by putting the values of 

 i/*' 6 " in (3). Now the equations (4), in this case, have only k linearly 

 independent solutions, since there are only k elementary divisors corre- 

 sponding to Ti , and therefore at least one ki\\ minor in (5) is not zero 

 when r =. r l . If, then, we have any other solution of (3) of the form : 



Vi= C t & (i = l, 2, . ..»), 



the constants C ; must be linear combinations of the k sets of constants 

 A; „ . We have then the following result : 



