298 



we shall find that, of the three quantities, s 0 , s l} s 2 , two are always 

 equal, and the third differs from them hy unity. 



I mentioned at the same time that I had arrived at theorems, ana- 

 logous, but less elegantly expressed, by summing the series formed by 

 taking every fourth or fifth coefficient, and so on, in the binomial deve- 

 lopment ; and I asked Sir "William E. Hamilton whether he remembered 

 to have seen these theorems stated anywhere. I thought it likely that 

 the well-known elementary theorem respecting the equality of the sums 

 of the alternate coefficients in the binomial development would have sug- 

 gested research in this direction. In a note, written on the day on which 

 he received mine, Sir "William stated that my theorem was new to him, 

 and that he had proved it by the help of imaginaries and determinants. 

 The following day he wrote again to me, furnishing me with the fol- 

 lowing more precise statement of my theorem : — 



"Let v and iVbe the following (whole) functions of n, 



then iV, iVand N+ v are always the value of the three sums, if suitably 

 arranged ; and the singular sum is s 0 , or 8 lf or s 2 , according as n, or 

 n+l, or w + 2 is a multiple of 3." 



I communicated the following demonstration of my theorem to Sir 

 William, in a letter of the 29 th March : — 



Using the notation employed above, we know that 

 [n + l) r = n r -f- n r - u 

 (n + = n r .i + n r - 2 , 

 and (n+l) r - (n+l ) r _ x = n r - ?i r . 2 . 



Now, putting s r = + n r „ m -f n r + n,. Jrm + , 



s' r - + 0 + l),._ m + 0+ l) r +(n+ l), +m + , 



(m being any positive integer), 



we have, from equation (1), 



S f r — S'r_i = S r — 5,._ 2 



and, in the particular case under consideration, viz. m = 3, 



S 2~ S i~ S>2~ Sq , 



S i — s' 0 = Si — S 2 j 



S 'o ~ S 2 ~ S 0 ~ S l • 



Thus it appears that the differences of the quantities 

 equal in magnitude, but of opposite signs to those of s l} s 2 , s 0 ; and ' 

 we form these differences for successive values of n, they will arrang 

 themselves in a cycle of six. Thus, if 



