TRANSACTIONS OF SECTION A. 527 



Hence if any column of the completed scheme be read upwards from the last row 

 «,, a T . . . , a n , the rule for the suffixes is to place above a ( the letter «. where j — 2i 



unless 2% > n, and j - In + 1 — 2 i in case 2i > n, in which case 2n + \ — 2i~ra. 



As another example suppose m = 7 ; the scheme, writing only the suffixes, is then 



12 3 4 5 6 7 



and there are four rearrangements ; the column of 5 remains unaltered and two 

 columns consist of 3, 6 repeated. The table thus gives a partition of 7 into 4 + 2 + 1 , 

 in which each number after 4 divides 4. 



In general we can prove that the number of substitution in the group is the lelht 

 number r ivhieh is such that one of the two numbers 2'— 1, 2'+ 1 divides by 2n+ 1. 



For suppose 2'' + e = (2n + 1)M, where e is + 1 or — 1 ; let 2 'be the least power of 

 2 greater than the odd number M, 2 3 the least power of 2 greater than the odd 

 number 2 ' — M, and so on ; thence we can write 



or 



M = 2 X '-2* 2 + 2' 13 - .. -2 V '+1, when e = +1 

 M = 2 X '-2 X2 + 2 X3 - .. +2 M -1, when «=-l, 



with X,>\ 2 >\ 3 . . ., the identity 1 = 2 — 1 being used if necessary to give the 

 required last term. It can then be seen, if N denote 2n + l, that the numbers 



H 2s = 2"^-i{(2 :V '-2 X - , + . . +2 M " 1 )N-2'| 



H 2 , +1 = 2^ S |_ (2 X '_2 X V . . . -2 XaS )N + 2'-} 

 are positive integers such that 



2Vi" X PH i , + l=N=2Vi-V 1 H p +l. 

 Beginning now with the number 1, apply the rule obtained above, j = 2i when 

 2i<w, j — N — 2i when 2i> ?i; it will then be found that we obtain in succession 



the numbers 



1, 2, 2 2 , 2 3 , . . . gr-v-i, 



N-2'-\ 2(N-2'- x .), . . . 2*.- x a- 1 (N-2'- x .), 



H 3 , 2H 2^->a-i(H 3 ), 



H 4 , 2H 4 , . . . 2^-^-ifl 4 , 



and so on, the last element of the last row being n itself. 



The number of these numbers is r ; though beginning with 1 and ending with n 

 they do not generally consist of all the numbers from 1 to n. In particular if N be 

 a composite number, no divisor of N will occur in the series. Let rfbea divisor of 

 N, and N = df; the equation 2'' + e = NM is the same as 2'' + e =fdM =/K, say. Sup- 

 pose r, is the least number such that one of the two numbers 2'"» + 1, 2 r > - 1 divides 

 by/, say 2'> + 6, =/M„ so that r^r. We may then form a cycle, 1, 2, 4, ... !(/"- !)> 

 of r, numbers, by the rule that after any number <r of this cycle shall follow r — 2a 

 so long as 2<r^A(/— 1), and the number /— 2<r when 2<r>i(/— 1) ; if we put i = da, 



j = dr, these rules are equivalent toj = 2i orj = N — 2i ; applied to d, they give a cycle 

 d, 2d, id,. . . ., ending with g(N — d), consisting of r, numbers formed by the same 



rule as was originally used. If we put r = ^r, — X, where 0^\<r„ the congruence 

 2 r > = ± 1 (mod./) gives, since 2'= ±1 (mod./), also 2*= ± 1 (mod./) and hence A. = 0. 

 Thus r divides by r,. Further, 2 r '=±l (mod./) gives 2'=( ± l) r ' r > (mod./), and 

 ( i l)'7''i can be — 1 only if ± 1 is really — 1 and r)r t is odd ; this does arise. 



Suppose next that p is any number <n which is not a divisor of N and does not 

 occur in the original cycle 1, 2, 4, . . ., . Let t be the least number such that one 

 of the two numbers (2' ± ])jj is divisible by N, and put 



(2 ( + e)^ = (2"'-2"=+ . . . +e)N, 



M m 3 



