398 Froceedings of the Royal Society 
Similarly d = , . . . , - ( - 1)*K(2'2 , . • . , ^ 2 )^ * 
, . . . 
> 22) + ' 
I2 
1 . 
.. 0 
0 
0 
-1 
? 3 * 
, . 0 
0 
0 
0 
0 . 
• • I2 
1 
0 
0 
0 . 
..-1 
h 
c 
0 
0 . 
., 0 -^ 
1 
M 
But this determinant is the complementary minor of the element in 
the place (1, 1) of the former determinant: hence by the funda- 
mental application of continuants, 
2« ^ 1 T 
^^i + - 1_ 
+ 2, + i 
• • . 22 ) 
M 
Our theorem thus is — - 
If sja‘^-)rd== 
aF i 
+ ^2 + 
* 
lii_ 
• • + ^'2 d" S'! “t 2 a 4- • • • 
* 
then 
2^^ 1 1 i (~l)‘"Kfa,...,g, ) 
d ^ q^-\ H g-g + S'! + M 
where I is the number of terms of the cycle and m some integer or 
zero. 
When I is odd and ^{q ^ » • • • ? ^ 2 ) certain special values, a 
converse of this is possible : and it is thus that the theorems 
in §§ 9, 12 originate. 
Additional Note. 
(Ordered by the Coimcil to be printed in sequence to Mr Muir’s former Note.) 
On the 3d April 1883, M. Catalan presented to the Belgian 
Boyal Academy a paper on continued fractions and certain series. 
