order 660 as a group of linear substitutions on 5 symbols 249 



is a solution of the above equations, the other solutions being 

 aoj', hoi'^, where w is any primitive fifth root of unity. With these 

 values of a and 6, the operation A, when expressed in terms of 

 the original variables is 



5a:,' = 2a>^-^|/ = lo + ^'«^4 + ^'^'^^3 "r oj^'b-% + oj^'a-% 



3 



+ a^i (1 + aoj^-^ + 66o2^-3 + 6-W-2 + a-^o;*^-!) 

 " + CCg (1 + «^'"^ + ^^^'"^ + 6-iaj3»-4 + a-ia;4^-2) 



+ ^3 (1 + aC0*-2 + 6a>2^-4 + 6-lco3i-l + a-l,o4i-3) 



+ CC^ (1 + aa;^-i + 6a;2^-2 + b'^co^'-^ + a-^o^^^-^) 

 (i = 0, 1, 2, 3, 4). 

 On entering the values of a, b, a'^, 6-^, it is found that there are 

 only five distinct coefficients, viz. 



-T§,(2Ao + 4A2 + 3A3 + 2A4), 

 and those derived from this by cyclical permutation of indices. 

 Finally, writing 



^, = 2A, + 4A,+2 + 3A,+3 + 2A,+4, 



the substitution A is 



- ll^o' = /3o»o + ^3^1 + te + ^4=»3 + ^22^4. 



- lla;/ = ^3*0 + ^iXi + ^4^2 + ^2«3 + ^02^4. 



- lla;2' = iSiCCo + ^^^i + ^2^2 + ^0^3 + ^s^i^ 



- UX^' = ^^Xq + 182CC1 + ^0^2 + te + ^1^4> 



- 11X4' - 182X0 + iSo^i + te + ^1^3 + ^43^4- 



This substitution and P, viz. 

 Xq = axQ, x{ = a%, cca' = «%> ^3' = «^^3. 3^4' = a^aj^, 

 generate F. Since the invariant has rational coefficients, if it is 

 unchanged by a substitution T, it must necessarily be unchanged 

 by T derived from T by changing the sign of a/- 1 in each of the 

 coefficients of T. It follows that F and F consist of the same set 

 of substitutions, the correspondence between the substitutions and 

 the operations of the abstract group being distinct for the two 

 representations. 



