(183) 



FURTHER NOTE ON FACTORIZABLE CONTINUANTS. 



By Thomas Muie, LL.D. 



(Read September 28, 1904.) 



1. The main result contained in the previous note * was a 

 generalisation embracing as special cases a theorem of Sylvester's 

 and a theorem of Painvin's, Sylvester's theorem for the sixth 

 order being 



a 1 ... . 



5 a 2 . . . 



. 4 a- 3 . . 



. . 3 a 4 .. 



. . . 2 a 5 



. . . . 1 a 



(a°- - l 2 ) (a 2 - 3 2 ) (a 2 - 5 2 ). 



The mode of treatment consisted in the removal of the factors, one 

 by one, in the sequence 



a + 5, a + 3, a + 1, a-1, « - 3, <x -5, 



each removal being followed by a lowering of the order of the 

 determinant. 



2. To this an alternative course has since been found which, 

 though not more effective in the matter of demonstration, has 

 unexpectedly led to a new theorem of greater interest than that 

 under consideration. Instead of removing one factor, advantage 

 is taken of the fact that the continuant in question is centro- 

 symmetric, and therefore is expressible as the product of two 

 determinants of the third order, viz., in the form 



a 



5 a 



. 4 a + S 



a 



4 a 2 

 . 5 a-B 



* Trans. S. A. Phil. Soc, xv. (1904), pp. 29-33. 



