299 



s 2 — Si — Aoj Si — Sq — A 2 , 5 0 — (? 2 — A], 

 we might form the following Table. 



n 



A 0 



A 2 



Ai 



\ 1 



- 1 



0 



1 



2 



- 1 



1 



o 



3 



0 



1 



- 1 



4 



1 



0 



-1 



5 



1 



-1 



0 



6 



0 



- 1 



1 



7 



-] 



0 



1 



8 



-1 



1 



0 



Combining this result with the well-known theorem — 



s 0 + «i + s 2 = n 0 + n x + n. 2 + = 2 M , 



we arrive at formulae for s 0 , $ l} s 2 . 



A couple of days later, I communicated to Sir "William Hamilton 

 my statement and proof of the corresponding theorem respecting the four 

 sums obtained by adding every fourth binomial coefficient. 



The theorem is as follows : — ■ 



" "Writing v = (- 1)' where i is any positive integer, 



If n is of the form 4.i, 



.'' n-2 



s 0 = 2- 2 + v2 T , 

 s 2 = 2»" 2 - v 2 T , 

 If n is of the form 4i + 1, 



