1904— 5. ] Continuants whose Main Diagonal is Univarial. 507 
Continuants whose Main Diagonal is Univarial. 
By Thomas Muir, LL.D. 
(MS. received December 12, 1904. Read January 23, 1905.) 
( l ) In a recently written paper * dealing with a continuant first 
considered by Cayley, it was pointed out that the function in 
question owed its complicated law of development to peculiarities 
of specialisation, there being a much more general continuant 
governed by a simpler law. The theorem enunciated regarding 
the latter was : — If A r be written for the sum of all the v-ary pro- 
ducts formed from b x , b 2 , . . . , with the restriction that no two 
consecutive b’s shall appear in any single product, then 
(2) The curious fact has now to be noted that this theorem 
itself can be generalised with a minimum of change in the mode of 
expression by altering the 2nd, 4th, 6th, ... . diagonal-elements 
on the left into </> and writing 6cf> for 0 2 on the right, the resulting 
theorem being then formulated as follows ; — 
6 b x 
-10 \ 
= 6 n + A l 0 n -‘ 2 + A 2 0 m - 4 + . . . 
- 1 0 
n 
For example, when n = 6 the expansion is 
0 \ 
- 1 </> h 2 
-1 0 b 3 
- 1 <t> 
(I) 
n 
= (0cf>) m + A f0cf>y - 1 + A 2 (0</>) m ~ 2 + . . . 
=0 j ■+ | 
when n = 2m - 1 . 
when n = 2m, 
* See Messenger of Math. , xxxiv. p. 126. 
