190 Transactions of the South African Philosojyhical Society. 



s = 



There are at least two interesting special cases, viz., when 



1 %m — 1 



b, and when s = - — —b. With the former substitu- 



m - 1 

 tion we reach the identity 



m 



ct-f-r (m-l)/3 

 -mfi ct+|H T 



a + ^-H ±{m-2)fi 

 . -i(m+l)/3 a+|H 



a+|H i(M-3)/3 



2ra -1 



= (a + t + m - l./3) m (a - - + m/3)™- 1 , where H = (4m - 2)/3 ; (V.) 



and with the latter the identity 



a-f- (m-l)/3 . . . . 



a r — jo . . . 



-+/3 a (m-2)/3 . 



(ra+l)/3 a r-2/3 



-+2/3 a (m-3)/3 



m/3 



2w-l 



- (a + r+m-l./3)(a-r-m-2./3)(a + r+m-3./3) (VI.) 



which degenerates into the original on putting j3 — 2B, and 

 r = (2m - 1)B. 



8. On returning to the two main results, (II.) and (IV.), it is 

 readily seen that, although the two continuants seem to be functions 

 of four variables, viz., a, b, s, r, this is not really so, because the 

 right-hand members are expressible in terms of three variables only. 

 In the case of the even-ordered continuant these latter variables are 



a + b, s, r, and in the other case a + b + r, s + t b, s + 2r. 



m — jl 



Putting therefore r = 0, s = 0, a = - b in (II.), and s = - 2r, 

 5 = 2(m - l)r, a— -(2?^-l)r in (IV.), we obtain the correspond- 

 ing nil-factor continuants, that is to say, continuants whose matrices 

 may be added to the matrices of the continuants in (II.) and (IV.) 

 without affecting the identities. For example, knowing that the 

 continuant 



a + 4 3 . . 



5 a 2 . 



6 a 1 



. 7 a 



= (a+7)(a-5)(a + 3)(a-l), 





