(153) 



NOTES ON SEMICIKCULANTS. 



By Thomas Muir, LL.D. 



(Eead May 31, 1905.J 

 1. Some attention has been given to the eight-line determinant 



d e J g h 



a b c 



h a b c 



g h a b 



f g h a 



h g f e 



a h g f 



bahgfedc 



c b a h g f e d 



d :e f g 



c d e f 



b c d e 



d c b a 



e d c b 



but apparently with no satisfactory result. The evaluation of it 

 was set as a problem in the Educational Times for April, 1904, and 

 a supposed solution was given in the number for September of the 

 same year,* the result being that the value was found to be the 

 same as that of the ordinary circulant G (a, b, c, d, e, f, g, h). But 

 although the first four rows of the given determinant are the same 

 as those of the said circulant, the last four rows are really the first 

 four rows of the reverse circulant C (h, g, f, e, d, c, b, a). The matter 

 thus seems to call for more careful investigation. 



2. Let us first make a fresh examination of the ordinary circulant. 

 Performing on it the operations 



we find it equal to 



colj + col 3 + col 5 + col 7 , 

 col 2 + col 4 + col 6 + col 8 , 



iO 



£ 



C 



d 



e 



f 



9 



h 



£ 



(1) 



b 



c 



d 



e 



f 



9 



0) 



£ 



a 



b 



c 



d 



e 



f 



e 



IO 



h 



a 



b 



c 



d 



e 



0) 



£ 



9 



h 



a 



b 



c 



d 



£ 



U) 



f 



9 



h 



a 



b 



c 



it) 



£ 



e 



f 



9 



h 



a 



b 



£ 



ti) 



d 



e 



f 



9 



h 



a 



* Bee also Math, from Edzic. Times, (2) vii., p. 55. 



