374 Transactions of the South Africa^i Philosophical Society. 

 when X is put equal to 1 : we thus have Pascal's result 

 1 2 



1 



31 



1 



2! 





2 









1 



1 





1 







5! 



4! 





2! 



' 





1 



1 





1 



.... 2 



1 



•••• 2! 



1 

 "'• 31 



2 

 1 



{2n-3)l 



1 



(2n-4:)l 

 1 



(2' 



1 



{'2n-l)\ 

 1 



(27^-2)! 



1 



(2- 



^-4)! •• 

 1 



{2n)l 



{2n - 1) ! 



{2n-d)\ " 



= 



,(xv) 



from the second order onward. 



Utihsing then the fact learned from (VIII.), that every such result 

 has a companion, we obtain 



1 



1 



2! 







1 



1 



1 



3! 



4! 



2! 



1 



1 



1 



5! 



6! 



4! 



{2n - 3) ! (2w - 2) ! {2n - 4) ! 

 1 1_ 1 



{2n - 2) ! 



1 



(2n - 1) ! (2n) ! 

 1 1 



1 



2! • 



4! 2! 



1 . 



(2w-2)! (27^-l)! (2n-S)\ 3! 



which also holds from the second order onward. 



(xvi) 



5. Expressions similar to those given in (XI.) for 



and in (IV.) for 



|N,D,|, |N,D3|, ..., |N,DJ 



