301 



x 1 , in the expansion of (1 + x) n for an arbitrary x ; so that we have al- 

 ways n Q = 1, but n r =0 in each of the two cases, r < 0, r>n. 



2. Let p be any whole number > 0 ; and let the sum of all the coef- 

 ficients n m , for which m = r (mod. p), the value of n being given, be de- 

 noted by the symbol, 



■(p) 



8 



n, r i 



which thus represents, when n and p are given, a periodical function of 

 r, in the sense that 



Cp) ,G>) 



s = 



n, r n,r + tp } 



if t be any whole number (positive or negative). 



3. A fundamental property of the binomial coefficient n r is ex- 

 pressed by the equation, 



(n + l) r = n r + n r , 1 ; 

 from which follows at once this analogous equation in differences, 



Cp) ti>) CP) 

 S , = S + 8 ■ 

 n, r > n,r-l t 



with the p initial values, 



Cp) Cp) Cp) Cp) 



s o,o =" : l > s o i = 0, 5 0 , 2 = 0, ... « 0t p.\ = 0. 



4. Hence may be deduced the general expression, 



Cp) 



s„, r = p~ x 2ar*"(l + a;) n ; 



in which the summation is to be effected with respect to the p roots x, 

 of the binomial equation, 



£^-1=0. 



5. The summand term, 



usually involves imaginaries, which must however disappear in the 

 result ; and thus the general expression for the partial sum, s, may be 

 reduced to the real and trigonometrical form, 



s = p' 1 2 (2 cos — cos — — , 



^ p J p 



with the verification that 



m7r\* . m(n-2r)7r 



0 = 2 (2 cos — sin — : 



P 1 P 



It. I. A. PEOC. — VOL. IX. 2 S 



