1900-1901.] Dr Muir on Pairs of Consecutive Integers . 
265 
(a + b)(b + c) - 2be = ab + b 2 + ac-b(2b + a ) , 
= ac- b 2 , 
= ± i ; 
and that the well-known identity 
{cfi + iab + Zbrf+^ab + W} 1 = { a 2 + 4a& + 56 2 } 2 
gives 
{(a + 6)(J + e)} 2 + {|fc} 2 = (& 2 + c 2 ) 2 . 
(5) As x/2 = 1 + J + , it follows that q r , q r+1 are the 
same as b, c ; and as a law of continuants gives 
( 1 , 2 , 2 , 2 ,...) = ( 2 , 2 , 2 ,...) + ( 2 , 2 ,...) 
we have 
Pr = g r +sv-i = & + «» 
and p r+1 = q r+1 + q r = c + b. 
The identity of (1) and (2) is thus apparent. 
(6) The curious proposition which forms the basis of Mr 
' 
Christie’s improvement is to the effect that 
2o + 2j + 2 2 + ... + 2 2r _ 1 
= 2'2 r _ 1 ‘2 r or (2 r _ 1 + 2 r )(2 r _ 2 + 2 r _ 1 )-l when r is even, 
= 2* 2 r _i 2 r —l or (2,_! + 2 r )(2 r _ 2 + 2 r-1 ) when r is odd. 
Tor the purposes of proof suppose the proposition to hold for r=2s, 
— -that is, suppose 
2 0 + 2 1 + 2 2 + . . . + 24 S _! = 2 , 2 2s _ 1 *2 2s . 
Trom this we have of course 
2 0 +2i + 2 2 + . . . + 2 4s _! = 2 , 2 2< _ 1 , 2 2s + 2 4s + 2 4s+1 , 
= 2-2 2s _ 1 # 2 2s + (2\ s + 2 2s+1 ) + 2 2s (2 2s+1 + 2 2s _!) , 
= 2 2£ _ 1 {2*2 2s + 2 2s _ x } + 2\ s + 2 2s ( 2 2s+1 4- 2 2s _ 1 ), 
= 2 2s _ 1 * 2 2s+1 + 2 2 /2 2s+1 + 2| s + 2 2s * 2 2s _! , 
= (2 2s +i + 2 2s )( 2 2s _ 1 + 2 2s ) ; 
and this we know otherwise (§ 3, footnote) 
= ^‘^2s+1^2s ~~ 1* 
