1904-5.] Dr Muir on the Theory of Continuants. 
667 
%l%2 ~ ~ 
K 
M 
iV 
where 
K = 
2/1 y. 2 
^2^2 2 l) 
a;, iCr 
a *2 
L — yfz 2 — z 2 
M = ^ 2 (^ 2 — ^ 1 ) 
2A 2/2 
2o 
/ \ ( / \ I ^2 
(2/3 " 2/2) I W 2 - h) ~ «2 L - 
1 2 
j- — (x 3 a? 2 )L 
N — (^ 2 zf x 2 (z 2 z i) 
'i *2 
j" — ^ 2(^3 “ ^ 2 ) ( 2 2 S l) ’ 
Unfortunately, the whole of this is almost hopelessly incorrect.* 
Muir, Th. (1874, Jan.). 
[Continuants : a new special class of determinants. Proceedings 
R. Soc. Edinburgh , viii. pp. 229-236.] 
Still another discoverer, twenty years after the publication of 
Sylvester’s first paper on the subject ! Fortunately the ‘ dis- 
coveries ’ are not expounded at inordinate length, the paper 
* The deletion of z 2 where it appears in K and elsewhere as the cofactor of 
a two-line determinant would effect an amendment of the second part of the 
work, the values of K, L, M, N being then reducible to z 2 (x 2 -x 1 ) | y x z 2 | , 
^ 2 ( 2/2 “ t/i) > z . 2 (z 2 - Zi)|l x 2 y 3 \, 2_>|1 x. 2 z 3 \; but there is an error in the first part 
as well. 
The four-line determinant, which is the denominator of the fraction sought 
above to be expressed as a continued fraction, is really transformable into 
I 1 Vl Z 2 I Z 2~ Z 1 
I X 0Vl Z 2 ! I x l z 2 I X 2 ~X 1 
I #1*2 I Va -Vi 
z 2 - z 1 
I 1 X 2Vz I 
I 1 X 2 Z Z I 
-r | y x z 2 I {z 2 - z x ) {x 2 - x x ) 
and the four-line determinant, which is the numerator, is obtainable from 
this by putting 0 for x 0 , y 0 , z 0 , — a substitution which only changes the first 
two elements of the first column into | y x z 2 | , 0 . The continued fraction 
thus is 
I Vl Z 2 I 
I I v&2 1 
( z 2 ~ gj) 1 x oI/l z 2 1 
I X 1 Z 2 I 
(x 2 - X x ) 
I y^ z 2 1 
(z 2 -z x ) I 1 x 2 y s ; 
| 1 x 2 z 3 | 
2/2 - 2/i 
