6 REPORT — 1856. 



■where it must be observed that P and its various powers have no meaning, 

 until the expansion has been effected. 



7. Ex. "Notation." 



The number of permutations in each case is contained in the expansion of 



or 



\{ (1 +P.r)«+ 3(H-Pj:)H 3(1 + Pa;)^+ (1 +Pa:)=} 



= \+bVx+ — P-x^ + 1 6P^x + ^ PV + ^^ P^-' + H Fj« + PV + ^ x\ 

 2 4 4 o o 



In this case, therefore, 



8Pi = 5: 8P2=23: ^V^=^d& : ,P,= 354 : 8P5=1110= 8?,= 2790 ; 



hP, = 5040: 8Pg=5040. 



To test these results, examine gPg. 



There are five different letters, n, o, t, a, i, whose permutations taken three 

 together ^=- 60. 



There are twelve groups of the form " nno," each of which may be per- 

 muted three times, or there are thirty-six permutations of this form. In all 

 60-1-36=96. 



8. It is presumed that a general method is preferable to the tentative pro- 

 cess, -which requires considerable acuteness in detecting the several groups, 

 and leaves a liability to error after all. Hence it is hoped that this theorem, 

 which supplies a desideratum in every-day algebra, may be worthy of the 

 attention of the Meeting. 



II. A particular Class of Congruences. 



1 . If S,n denote 



i'»-^2'"-h3'"-i- .. -^ip-\y\ 



where ^ is a prime number, 



2,n = o (mod.jB.)j 

 unless Tn=-r(^p—\), when 



2r(p-l) = (;^-l)- 



2, If a, b, c, d denote four of the series 1, 2, 3, . . /) — 1, 



2(aP-»6P-icP-')^(^_l); •Z{aP-H''-'cP-UP-')=\ J^ ^^ 



1. If j) is prime, the congruence 



x—\ .x — 2,... x—p+\—{3^~^ — \)^,o (mod.;?.) 



has;? — 1 roots 1, 2, 3, . . . ./?— 1 : and since this congruence is only of the 

 ( jj — 2)th degree in x, the coefficients of the several powers of x are sepa- 

 rately congruous to p. Hence we have 



where Sj denotes the sum of the roots, 



s., taken two and two, 



Sp-i their product. 



The above paragraph contains Serret's demonstration of Wilson's theorem. 



I 



