384 Proceedings of Royal Society of Edinburgh. [sess* * 
well-known property of continuants (v. Math. Gazette , ii. pp. 58-59) 
(Pn ~ R-l ) 2 + fyn-lQn = (Pn + <ln- if “ 4( - 1 ) w , 
and therefore, if w be odd and e be even, 
+ 1 ) 2 + 4 or (^ e + g 6 _i) 2 -4. 
Also as regards the square roots of these forms it is known that 
VW + 4 - 
* * 
1 1 1 
V(2M + l) 2 + 4 = (2M + 1) + 
V ( 2M + 4) 2 -4 = (2M + 3) + - +m + t 
M+1 + 1 + 4M + 2 + - • • 
* * 
111 1 
+ 4M + 6 + • • ■ 
* 
V(2M + 3) 2 -4 = (2M + 2) + 
11111 1 
1+M + 2 + M+1 + 4M + 4 + - • - 
* 
* 
Consequently, taking the first case only, viz., where Po, + - 1 is- 
even, we have 
\/(i\> + g w -i) 2 + 4 
{Pu + q«>- 1 ) + 
1 
i(Po> + q co-i) 
* 
l 
+ 2(^> w + _ i) + • • • 
* 
whence, by addition of y> w - - 1 to both sides, 
2 I flq + — , , ~ , \ = 2 pi* + 1 
( * & 2 + a w + • * ’ ) Up<o+q co-l 
* 
and finally 
i(p a + q*-l) + 2(p a + q„„ 1 ) + " 
* * 
fli + 
* a 2 
+ • 
I l 
+ a w H j 
= Pu> + 
1 
p<o + qu-i + - • • 
* 
For example, 
(VI) 
7 
1 + 1 - 1 
* 2 + 3 + 
10 + 12 + 
_1_ l_ 
12 + 12 + .. • 
* 
Even although we do not take an infinite number of periods, the 
identity still holds, provided that the same number of periods be 
taken on both sides ; thus — 
