294 REV. T. p. KIRKMAN ON THE THEORY OF 



20. Let gQ=S{pi + c) (mod. (N-i)) be the group of 

 (N-i)R;v_i substitutions made with N-i elements of 

 theorem D and of Art. (15), when N- 1 is prime, in which 



case 



(N-i)R.,_,-(N-i)(N-2). 



Let the final element N be added to every one of the 



(N- i)(N- 2) substitutions of ^q- This gives a group made 



with N elements of the order (N- i)(N-2), 



N 

 ff = S{pi + c) (mod. (N-i)) + - 



where i is any number < N, p is any number > o and 

 <N- 1, and c is any number <N, '>o. 



The substitutions of g are formed by writing jn + c for 

 i, when i<N, and N for i = 'N. 



This 



with the N - 1 derived groups 



completes a group of N(N- i)(N-2) substitutions; if 



W" = tP "' + 1 } mod. (iV-1) I J: 1 ^ 



'~ {/3-+iUd.(iv-i) "^N+1^ 

 and if, for r> 1, 



_ { /3-a+^) (i-r)+rU,. ,j,_,^ , N r 



where /3 is any primitive root of (N-i), and where the 

 numerators and denominators of the first fractions are 

 residues estimated to modulus (N- 1). Every one of the 

 elements < N can be represented by a finite value of x in 

 the first denominator of W^ except the element r, and the 

 rest of the expression gives the substitutions to be made 

 for r and for N. 



If we admit an infinite value of a^, the value r is not 

 excluded from those represented in the terms of the first 

 fraction in "¥,., and the substitution retains its definite 

 character. 



