386 
Proceedings of the Royal Society 
we transform it into 
• 
* 
* 
* 
• 
• 
* 
* 
• 
• 
* 
* 
* 
• 
* 
*■ 
-+• 
* 
• 
* 
*• 
+ 
* 
• 
• 
* 
# 
• 
• 
* 
* 
* 
• 
* 
* 
* 
• 
* 
* 
* 
• 
• 
* 
* 
• 
• 
* 
* 
* 
• 
the middle term of which*becomes by transposition of the first two 
rows, and the subsequent transposition of the first two columns, 
• * # # 
• • * * 
* * 
Consequently we have 
*(5) = X,(4)+x/4) + X s W. 
and it is easily seen that a similar transformation is possible in 
every case, giving 
*0) = XiO - !) + xl n “ !) + X*( w - 1) + . . . . + x«- a(w - 1) . 
Expressing x 2 > Xa> — terms of x 0 by means of wbat precedes, we 
have 
y( n ) = Xo( n ~ 1 ) + Xo( n - 2) + Xu( n - 3 ) 
+ XoO “ !) + XoO “ 2) + XoO “ 3 ) + XoO “ 4 ) + XoO - 5 ) 
+ XoO - 1) + XoO - 2 ) + XoO - 3 ) + XoO - 4) + XoO - 5 ) 
+ XoO - 1) +XoO - 2 ) + XoO “ 3 ) » 
