384 REV. T. P, KIRKMAN ON THE THEORY OF 



the -(N- i)**^ order formed on the partition 

 N=(-N^)2+i.i=Aa + BZ». 

 To each of these can be added -(N- i) didymous radi- 

 cals, and this can be done in -(N-i) different ways, by 



theorem G, (26); -(N-i) = yA", (25). 

 2 ft 



One of the groups [h), hj^, which has N final undis- 

 turbed, is composed of -(N- 1) substitutions 



N- 1 



3 systems of did 



2 "^ 



N- 1 

 and any one of its systems of didymous radicals is 



where 



(/,i( = )i-i + A(i-i-i*^^-^0 (mod.N) 



if A be determined by the condition 



(pi = c, (mod. N), 

 c being any number < N which is no power of ^. 

 91. It is only necessary to prove here that 



{cf>in=)i, 



or that (pi is a square root of unity, whence it readily fol- 

 lows that 



(f>{8'H)(j>{/3H){ = )l3H, (mod. N), 

 one of the substitutions of ^^. 



The former of these equations affirms the congruence 

 i = (i-i + A(i-i - ii(^-3)))-i + A(^-l + A(i-' - i^(^-3)))-i 



- A(i-1 + A(i-^ - ii(^-3)))i(^-3) ; 



or that 



i{i-^ + A{i-^ - i«^-3))) = 1 + A - A(i-i + A(i-i - ii(^-3)))4(^-i), 



which is 



1 + A - A^•*f^■-l) = 1 + A - A(i-i + A(i-^ - i«^-3)))i(iv-i)^ 

 This is true on condition that 



