356 BEV. T. p. KIRKMAN ON THE THEORY OF 



where I venture to introduce the new symbol [ = ) of tac- 

 tical equality, to be employed when we equate a substitu- 

 tion C to the product of its factors, as 

 C(=)AB, or C{ = )V^, or AB(=)P™ or CD( = )AB, &c. 



For it will be difficult to avoid confusion, if we continue 

 to employ the same symbol, =, in the tactical proposition 



{pi + c){qi + d)=A, 

 when we mean by A the substitution pqi+pd+c, and in 

 the algebraic proposition 



{pi-Jt-c){qi + d)=B, or ^B, 

 when we mean by B the number pqi^ +pdi + qci + cd. 



It is above evident, that, whatever p, m, or c may be, 



{pi + c)™( = )fH +^-^ c ; 

 p— 1 



whence, if p, being prime to the composite number N, is 



also a primitive root of a certain congruence 



y=i (mod. N), (a) 



for a value of m which is a divisor > i of N, there is a 



certain number of powers of pi + c of the form 



(joi + c)™"(=)2+^'-— — c(=)i + e, (mod. N). 



That pi + Ci may be of the W'' order, when c^ is prime to 

 N, it is required, and it suffices, in addition to the condi- 

 tions {a), that the congruence 



2)* — 1 



~ ^0 (mod. N), (that is, e = o) 



be true for no value of ^<N. When these conditions 

 are fulfilled, we have always, for all values of c> o < N, 



(joi+c)""'(=)i+ ^ _ c{^i + e 



a power of 6% of 



6'( = )i+i( = )234..Ni, 

 and 



{pi + Ci)'^'^{=)i, (Ci prime to N), 

 when m'/^^N, and only when mn^'N. 



7 1 . When c^ is = i , or prime to N, pi + Cy is always of 



