300 



If n is of the form 4* + 2, 



*0 = 



2«-2 j 



n- 2 



8 X = 





S 2 = 



n- 2 



S 3 = 



2«-2 _ v 2~; 



If w is of the form 4« + 3, 



n -3 



s 0 = 2 n - 2 - v 2~ T , 

 Si = 2 n ' 2 + i/ 2~, 



71-3 



s 2 = 2"" 2 + v 2~ 



n-3 



s 3 = 2»- 2 - v 2~ T . " 

 The proof of this rests upon the equations 



S 3 — S r 2 = 



*3 





5 2 — S 'l ~ 



S 2 





8 1 ~~ s 'o = 



*1 



-> 



8 o - S'a = 



*0 





combined with s 0 + s x + s 2 + s 3 + 2". 



Though the theorems which I have now stated or indicated are not 

 devoid of interest, I should hardly have brought them under the notice 

 of the Academy if they had not led Sir William E. Hamilton to discuss 

 the more general question treated of in the Note appended to this paper. 

 It is at his suggestion that I have communicated the substance of the 

 letter which I addressed to him on this subject. 



I may be allowed to add, that the first theorem stated in this paper 

 was suggested by the investigation of a very simple geometrical problem, 

 and that I have found that it admits of being very curiously illustrated 

 by means of my theory of algebraic triplets. 



Extract from a recent Manuscript Investigation^ suggested by a Theorem 

 o/Dean Gbaves, which was contained in a Letter received ly me a 

 week ago. 



1 . Let n n for any whole value not less than zero of n, and for any whole 

 value of r, be defined to be the (always whole) coefficient of the power 



