384 
Proceedings of the Royal Society 
Beginning with the first of the x forms we see it to be trans- 
formable into 
* * *• 
* * * 
*■ * 
* *• 
* *• * 
+ 
* * 
* 
* #• 
* * * 
and the second term of this may in like manner be changed into 
• • * * 
*■ *■ 
* * 
+ 
* * 
and the second term of this again into 
+ 
the second term of which is zero. Hence we have the result, 
Xo (w) = ^(n) + V(n - 1) + - 2) + . . . + tf(3), ... (a) 
Of the other x forms it is clear to begin with that 
Xl Xn — i > X2 Xn — 2 5 • • • • 
and treating the distinct cases as we have treated x 0 > we equally 
readily see that 
XiO) = XoM + X<k n - !) + XoO - 2 ) 
X*( n ) = XoO) + Xo( n - !) + XiO - 2) 
XsO) = XoO) + XoO - !) + X 2 O - 2) 
X»-aO) = XoO) + XoO - !) + XiO - 2) 
X«-lO) = XoO) + XoO “ !) + XoO - 2) 
