GROUPS AND MANY-VALUED FUNCTIONS. 389 



the circle of differences 



b — a, c — b, d-c, e — <?• • 

 is exactly the circle of differences 



b'— a' , c'- b', d'- c' , e'- d' • • 

 begun at some other point than b'- a' . 



We have only to form the N powers of the substitution 

 \ by writing under i in unity the vertical circle ^i, of 

 which the first element is always i, whatever c may be, 

 and then completing the same vertical circle under every 

 element of unity. 



If there be no power of A, whose circle of differences is 

 that of ^i, the group sought has no existence. 



If there be such power or powers, we have found <^i 

 consistent with equation Q, and as all the unknowns of Q, 

 in the right member can be determined in terms of B and 

 C, that is of A in ^i, we have proof that the group exists, 

 for the system of didymous radicals given with ^i. 



95. It is easy to prove by this method that no such 



group of -(N+ 1) N'(N- 1) exists for N=i3 or N=i7 



or N=i9. 



The substitutions ^i which give the sought groups are 



for N = 7, 



(f)i=-i-^ + 2p= 1462537, 

 and 



<^^=5^-l^- 3^2= 1432657; 



and for N=ii, 



^z = 32-1 - 2i*= 1843907256a, 

 and 



^i = 5^-1 -4^*= 1043976852a. 



The four groups thus found of the order '-(N+i)N' 



(N-i) may be thus expressed as products, denoting by 

 {H}^ the group of powers of the substitution H of the 

 m^"' order. 



{234567i},{i357246}3{i426735}4{i462537}2=F 

 {234567i}7{i357246}3{i347652Mi432657}2=r' 



