1904-5.] Dr Muir on the Theory of Continuants. 
659 
employed in dealing with known properties of continued frac- 
tions. 
The third section opens with the identity 
«o 1 
A a-. 1 
% 1 
1 A 4 1 
11 
b n a„ 
1 A 0 
• 2 2 
• -2 
a n- 1 1 
A n—l 1 
b n a n 
1 K 
where A-^c^-— , 
and generally 
a °i a b 2 
K- a 2 . A 3= a 3 6 1&3 
A r —a r - 
b r _ 1 b r _f r _ i . . . . 
t)f) r _cf r _^ . . . . 
This is reached by a rather curious process, of which only the first 
two steps need be indicated.* They are, in the case of the fifth 
order, — 
«0 1 . . 
a 0 1 
\ a 1 1 
t— H 
to 
tP 
b 2 a 2 1 
h «3 1 
b 3 a z 1 
a t 
1 «4 
h 
a o 1 ... 
11 
“0 1 
b ± a 1 1 . 
& 4 a 4 1 
b 2 a 2 1 
& 2 a. 2 1 
1 1 
1 a 3 1 
h 
J h 
1 a 4 
1 “A 
63 & 4 
1 
* The transformation is probably most readily effected by dividing the 2 nd, 
3rd, 4th, .... rows by 
\ , b 2 , b-J) 3 , & 2 &4 , b x b 3& 5 , & 2 & 4& 6 , > 
respectively, and multiplying the 3rd, 4th, 5th, .... columns by the same, 
the most instructive order of performing these operations being that in which 
each division is immediately followed by the corresponding multiplication. As 
the number of the divisions is necessarily one more than the number of multi- 
plications, the reason for the outside factor b n b n - 2 4 .... on the left mem- 
ber of the identity is also thus made apparent. 
