1900-1901.] Dr Muir on Pairs of Consecutive Integers. 
267 
x = J(2 2r + 2 2r -i - 1), y = 2 2rJ where 2 r stands for the continuant 
(2, 2, 2, . . . ) of the i th order. 
Apart from all that precedes this can be proved in a line or two. 
Dor, by substitution, 
x 1 + (x + lf = J(2 2r + 2 2r _ 1 - l) 2 + J(2 2j . + 2 2 r-1 + l) 2 , 
= + J + 2 2r 2 2r _ 1 , 
— i^2r + 2^r-l(^2r-l + 2 ' 2 2r ) + \ , 
= h%lr + i( 2 2r-l ‘2 2r+1 + 1) , 
= J2 2 r + I2 2 r = 2 2r , 
= y 2 - 
It is scarcely possible to think of the whole matter beings put 
more simply or in shorter compass than this. 
