232 Proceedings of the Royal Society 
and from this, with the help of (III.), we can prove that 
K (... a, &,0,0,e, /,...) = K(...a, b,e,f ,...) . (VIII.), 
and so, generally, when the number of consecutive zero elements is 
even. 
12. Using the ordinary process of finding the greatest common 
measure of two numbers, we may establish another special 
property of simple continuants, viz., that, whatever a lt a a , . . . 
may be, 
K On •••«*-!> O 
is prime to 
K (a„ <J 2 , . K(o a , a.), K (a, - 1, a„ . . . a,), and 
K (°i> »«> •• •“»-!)« 
13. When both diagonals of a continuant are the same when 
read backwards as when read forwards, it may be called symme- 
trical. 
In connection with simple symmetrical continuants, the follow- 
ing identities may be mentioned : — 
K (a L , a 2 , . .. o M _ 1} a n , a n _ x ,...o a ,a 1 ) = K(a 1 ,a s ,... a n _i) { K (a x , a s a w _ 2 ) 
+ K (a lt a 2J ...a n )} (IX.) 
K (a 1} a 2 , ... tf n , a n , a 2 , a A ) = K (a 1? a 2 , . . . a n _i) 2 
+ K (a 1? a 2 , . .• . a w ) 2 (X.) 
« w - 1 v..ft a ,a 1 )=K(aJ,o a ..^ l( ) 2 
. - K Ol,« 2J •■•«n- 2 ) 2 (XI.) 
K(a A , a 2 ,...a n _ 15 2o n , aj= 2K(a 1 , K (a lf a 2 ,...a n ) 
(XII.) 
Connection between Continuants and Continued 
Fractions. 
14. The value of the special study of this class of determinants 
lies in the fact that by means of them the convergents of a con- 
