TRANSACTIONS OF THE SECTIONS. 7 



Now observing the meaning of Sm . "we have from the theory of equations 

 the following relations between the symbols S and 5 in the equation 



J^-'-Sl xP-'^ + S^ X"-^- .... +Sp-i = 0. 



2^-s^ 2^ + 2 3^=0. 

 23-51 S2+S2 Si-3 ^3=0, 



Sp-i— *1 Sp_2 + S2 2p_3 — +(p—l)Sp-l =0. 



2p — Sj Sp-i + Sg 2p-2 +Sp-i Sj =0. 



S2P— 2 — *! 2I2P-3 + S2 ^2P— ■* — •• •• +fp-l Sp_i^O. 



Hence we establish the following congruences ; — 

 2i=Si ^ (mod. p.) 

 22=— 2^2 = 



2p_2 = 0. 



^p-i = —(p — ^)sp-i~p—l. 



22^-2= — Sp-1 Sp_i :^^ — 1. 



^r(p-l)=—Sp-l 'Z(;r-l)(p-l)=p—l. 



2. To prove the second proposition, we will premise the following con- 

 gruence : — 



1 .2.... r - ^ ^ ■'' 



if p is prime, according as r is even or odd. 



For ^^ — •• • -P — jg always an integer ; 



1.2.... r 



p— 1.»— 2 p—r + l.2....r. 



.•.^- i- ■ IS an integer, 



and is therefore a multiple oip, since p is a prime greater than any of the 

 factors of the denominator. 



3, S(a''~' )=i'— 1 = — 1> as has been proved above. 



_ (^-i)3_3(^-i)2+2(j>-i) _ 7^i .^2 .7:^ __ J 



— 1.2.3 ~ 1.2.3 ~ ■ 



_ (Sp-i)'' — 6S2P-2 • (Sp-i)^+8 . S3p_3 . 2p_i + 3(22p-2)''— 6S4p_4 

 ~ 1.2.3.4 



_ {p-\y-Q{p-\Y-\-\\{p-\y-^p-\) _ J^\ .J^ .J^ .J^A 



— 1,2.3.4 ~ 1.2.3.4 



= +1. 



