1904 - 5 .] Continuants whose Main Diagonal is Unwarial. 511 
(8) Proofs of the foregoing six theorems have been purposely 
omitted, because the modes of procedure followed in the case of 
the original ungeneralised theorems are applicable without altera- 
tion to the new theorems. In only one instance, that of (II), does 
previous work stand markedly in need of being supplemented. 
The first part of it, viz., where n is even, is best dealt with as 
follows : — 
<*w s = 
Ob,.... 
(f) l ... . 
“ 1 (f) h . 
1 . . . . 
• -i Ob,. . 
. - b 2 (f) 1 
. -1 (f) b 4 . 
. . 1 . . 
. . -10 b b 
. -b 4 (f) 1 
... .-!</> 
1 
= 
0<f> 4* b, b, — b-^bt) 
(f) 
- 1 - 1 0(f) 4- b 2 
4- b 3 b s - b z b 4 
</> 
-1 -1 0(f) + b 4 + b 5 b 5 
0(f> + b. 
- b,b 2 
-1 
0(f) 4- b 2 4 " b z — b z b 4 
- 1 0(f) 4 - b 4 4 - b b 
0<f> 4 - b. 
b. 
b 2 
0(f) 4 “ b 2 -}- b z b z 
b 4 0(f> 4 - & 4 4 - & 5 
Applying the same treatment to <E> when of odd order we obtain 
^7 = 
0(f) + b, 
h 
0(f> + b 2 + b z 
h 
0(f) 4 - &4 4 - b b 
b 
6 
+ (f), 
h 
0(f) + 1>q 
— a result interesting in itself, although not the form desired. 
Increasing each column by the column which immediately follows 
it, we have 
