385 
of Edinburgh , Session 1876-77. 
which, on eliminating Xd X 2 ) ••• fr° m the right-hand members, 
become 
XiO) = XoO) + Xo( n - !) + Xo(^ “ 2 ) 
XaO) = XoOO + Xo( n - !) + XoO - 2 ) + Xo( n - 3 ) + XoO - 4 ) 
Xs( n ) = XoO) + XoO - !) + XoO - 2 ) + XoO - 3 ) + XoO - 4 ) + XoO ”5) 
'+XoO“ 6 ) 
X«-aO) = XoO) + XoO - 1) + XoO - 2 ) + XoO “ 3 ) + XoO - 4 ) 
x«-i O) = XoO) + XoO - 1 ) + x<>0 - 2 ) • 
Here the second of the series of right-hand members has two terms 
more than the first, the third two terms more than the second, and 
60 on until we approach the middle of the series, when, if n be odd, 
the two middle right-hand members are found to be the same as 
the one preceding or the one following them, the whole four ending 
thus — 
• • • -+Xo( 5 )+Xo( 4 )+Xo( 3 ) > 
and if n be even, the one middle right-hand member is found to be 
greater by unity than the one preceding or the one following it, 
and to end thus — - 
. . . . + Xo(5) + Xo( 4 ) + Xo( 3 ) + 4 > 
the 1 arising from the fact that the above process of reduction, in 
the case of x * n ( n ), leads us finally, not to 
• * * 
• * * 
* * 
• * 
• • * 
+ 
, but to 
• • * 
+ 
• • 
# 
* 
* 
*•«•, t0 Xo( 3 )"h !• 
Eeturning now to the ^ form, and taking 
