188 Transactions of the South African Philosophical Society. 



factor and the factor immediately following it is 2a + 2/3-5, thus 

 giving for the sum of all the factors 



(a + b + r) + (m - 1) (2a + 2b - s) + (a + b-s- -), 

 i.e., m(2a + 2b-s) + 2 T . 



Similarly we note that the sum of the first and second diagonal 

 elements, and the sum of any other odd-numbered diagonal element 

 and the element immediately following it, is 2a + M, the result 

 being that the sum of all the diagonal elements is 



m(2o + M)+r, 



i.e., m{2a + 2b - s) + 2r. 



6. Putting s, which occurs only in the even-numbered factors of 

 (II.), equal to 2/3 + - - we deduce from (II.) the identity 



a+r b . . . 



; 1 VI -1 



0+—T a r . . . 



vi m 



m + 1 7 1 



T a 0— —t . . 



m m 



b + -r 



2 vi - 2 



a 



m vi 



m + 2 7 2 



t a - —t 



m vi 



= (a' + fr + r 



\ 



) • ( a - h - T+ i T ) 



. [a+6+r 



2_ 



) . (a-b-r+^r) 



\ 



77^ 



1 \ m / 



. [a+fr+r 



_4 r 



) • («-fc-r+V) 



\ 



77t 



/ V ™ / 



'2»i 



/ , 2/;/ - 2 \ / , , 2/// - 1 \ /TTT . 



. a + 5 + r- r . a-6-r+ -r . (III.) 



m J \ in 



Specialising still further by putting b = 2m - 1, and r = 2m, we come 

 back to the simple result (a) from which we started in § 2. 



