136 Proceedings of Royal Society of Edinburgh. [sess, 
then 
{Cf > i s • •• 5 ) ^m+1 ? • • • ) ^m+w) = (^l j • • • j a m){ a m+ \ ? • • • > ^m+n) 
+ ( a^ , ... } a m _f(a m+ 2 j • • • j ^m+w) ). 
— a possibly new result which he considers “the fundamental 
theorem in the theory of continued fractions.” This, he says, is 
an immediate consequence of the fact that (a 1} ... , a m+n ) can 
he expressed as a determinant, . all that is further necessary being 
the application of the “ well-known simple rule for the decomposi- 
tion of determinants.” Thus, e.g., the determinant 
a 1 
-1 b 1 
- 1 c 1 
-1 d 1 
-lei 
-1 / 
is obviously decomposable into 
a 1 x d 1 
-1 b 1 -1 e 
- 1 c - 1 
or into 
a 1 x cl 
-lb -Id 1 
- 1 e 
- 1 
or into 
ax b 1 
-1 cl 
-1 d 1 
-1 e 
-1 
+ a 1 x el 
i -i & -i /, 
/ 
+ 'a x d 1 
-lei 
1 -1 /, 
/ 
+ c 1 
-1 d 1 
-1 e l 
1 -1 /. 
/ 
Following this is what is called “ Corollary I. ” viz., 
1» » • • • > ^ m ) * (^2 J ^3> • • • J — (®2 ’ ^m) * 1 J .^2-» • • • » ^m+i) 
= ( - ) TO (« m+i am+i-i ... to ^ - 1 terms), 
its connection with the expression for the difference of two con- 
vergents being illustrated by the instances i — 1, 2, 3, 4, . . . 
