1904-5.] Dr Muir on the Theory of Continuants. 
669 
K 
\ • • • K-i \ = 
^1 J — ^2 ’ • • ' ’ ~ a i 
. . . b n _^ 
, (?2 fl, 
K(1 , ) $2 ) • • • * J + 1 , ®2 j • • • • 5 ^ra) 5 
K(...a, 6, c, 0, e,/, g ,...) = K(...a , b, c + e,f, g,...), 
K(...a, b, c, 0, 0, 0, e, /,...) = K(...a. &, c + e, 
K(... , a, 0, 0, e , §K(..., a, 6,.e, /,...) 
Then there follows the theorem that ivhatever a Y , a 2 , may be , 
the continuant K(a 15 a 2 . . . . , a„) is prime to K(a 15 a 2 , ... . a n _-^), 
K (a 2 ,...., a B ) , K(a 1 - 1 , a 2 , . . . . , a„) , K(a 1 , a 2 , . . . . , a n - 1) , 
the foundation for this being the ordinary division-process of finding 
the greatest common measure of two integers. 
Symmetric continuants are next defined and four properties 
stated, the two last of the four being 
a n 'K(a 1 ,..., a n _ Y , a n , a n _ x ,<..,%) = K (a 1 , . . . , a n f - K(a 1 , . . . , a n _^f 
K(a x , ..., a n _ Y , 2a n , a n _ Y , . . . , a x ) = 2K(a 1 , . . . , a„_ 1 )K(a 1 , . . . , a„) . 
The fundamental relation with continued fractions having been 
established, and thence the theorem 
/ K ( b i \ 
+ \ h h K b i = / V A > a i , a 2> , A 
a x + a 2 + • • • + a 2 4- a 1 + 2A + . . . rzf b 2 b 2 \ 
* * '^\a 1 a 2 . ... a 2 aj 
the deduction is made that 
XT ( b l b 2 ’ ‘ 
\A, a 1 , a 2 
• b 2 \ \ 
a 2 a Y A ) 
K 
2 2 
a 2 , . . . , a 2 a x 
K 
b \ b 2 
b 2 b Y \ b 2 b 2 b 1 
A, a lt a 2 ,...,a 2 , a x , 2 A , a 1 , a 2 ,....a 2 , a lf A 
K 
b 2 b l 
b l b 2 
a x , a 2 , . . . a 2 , a lt 2 A , a 1 , a 2 , ... ,a 2 , a Y 
Any other results given belong properly to the theory of con- 
tinued fractions.* 
* Some of these results and others are dealt with more fully in a pamphlet 
of 32 pages printed about the same time at the University Press in Glasgow, 
and entitled “ The Expression of a Quadratic Surd as a Continued Fraction.” 
