5 
the four expressions (a,x), (a 2 ,x), (a 3 ,x), (a 4 ,#) constitute a 
group of four, viz., the group 
1234 
2413 
3142 
4321 
wherein the numbers have reference to the suffices ol the 
xs. This group has a further peculiarity. Effect a cyclical 
permutation to the cycle 01234 in the group. This is 
effected by changing the four expressions into 
a(a,x ) j a 2 (a 2 ,x ) \ a 3 (a 3 ,x) ; a 4 (a 4 ,x ) 
respectively. Let 7 r be the product of the first four expres- 
sions, (ci,x), (a 2 ,x), &c. Then 'the product of the latter four, 
viz., a(a,,x), a 2 (a 2 ,x), &c. } IS equal to a.a 2 .a 3 .a 4 .7r = a 10 7r = 7r. Ill 
other words, n is unchanged by the cyclical substitution, 
and is at once a group of four and of five, and has — — 
or six values. Similarly the four expressions, 
j (a 2 #)} 5 , &c., are severally equal to the four, {a (a,#)} 5 
{a 2 (a 2 x)} 5 , &c., and any symmetric function, say, 
(a,x) 5 + ( a 2 ,X ) 5 + (a 3 ,#) 5 + (a 4 ,#) 5 , 
is also a group of five and of four. Let (a >) = Q,., and for 
a moment put 
This is a very different system from the system given by 
Mr. Kirkman, at p. 134 (op. cit.), which is 
x 0 = *2® + I(pf y + &c. = is* + i(s + H, j * + (fee. 
Mr. Kirkman says (ib.) that Hj is unaltered by the cyclical 
permutation of XqX^x^. But what of H 2 and H 3 and H 4 ? 
And if H 2 and H 3 and H 4 are not also unaltered by the 
same cyclical permutation, all connection with the theory 
of Lagrange or the method of symmetric products seems to 
be lost. I think that in our correspondence some ten 
years ago I said a great deal (or something, at all events) 
about the disruption of cycles. It might be worth while 
