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



The product is 



which is a substitution of the derived group 



viz. that in which 



c = c' + y'^~''c^ . 



In like manner the product 



V +v^c' )\ +11" c / 



+ y^'c / \ + y^c 



^(y''^+'+\e„,^z.n'-c')\_ 

 \ + y'^c' ) 



is a substitution of the derived group 



(2/(=i+^+i)K + c)) (^/(^^•+'^+i)(A„ + c)) . . (?/^^^-+^+^'(e™ + «„. + c)) • . , 



viz. that in which 



c = Ci + «/*■"■ ^■''^^c'. 

 45, We have demonstrated this theorem — 

 Theorem J. If ^ be any one of the numbers ABC- ■ of 



Art. (4.1), and 



2/ = E-i . 

 be a primitive root of 



w''=i (mod. M), 

 M being any one of ABC • • and at the same time a root 



primitive or not of 



x'^^i (mod. X), 



X being every one in turn of ABC- -, we can construct on 



the given partition of N [Art. 41) with this root E- 1^ 



{theorem C, 1 2) equivalent groups of the order h\ where e 

 is the multiplier of ^ in the partition of N ; these groups 

 being all of the form 



G + ©iG + e^G + . • + 0'-'G, 

 ivhere G is any one of the W groups of theorem C. 

 Take, as an example, 



N=io.i+8- 1+4'2 = 26 (^• = 40). 



