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 (II) 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 ^'s 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^, and suppose that 



(12) ri = ra = . . . = r^, 

 so that 



(13) ei + ^2 + . . . + e^ 



is the multiplicity of the root r^. 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 : 



Therefore the constants A^ ^ ^ are linearly independent solutions of the 

 equations (4) when r zzz 7\, as we readily see by putting the values of 

 y.."'^" 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 r^ , and therefore at least one kt\\ minor in (5) is not zero 

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



y.= C.x"-^ (» = i, 2, ...«), 



the constants G^ must be linear combinations of the k sets of constants 

 -4, ., «> • We have then the following result : 



