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the labour Mr. Cockle formerly expended upon epimetrics 
well enables him to appreciate. The Author next considers 
his symbol 0 as a rational and symmetric, but otherwise 
arbitrary, function of four other functions, one of the latter 
functions, again, being a rational, but otherwise arbitrary, 
function of four arbitrary symbols, and the remaining three 
functions being derived from it by the three phases of an 
interchange which, provided it be of the fourth degree, is 
otherwise arbitrary. He then expresses the results of all the 
binary interchanges that can be performed on & in terms of the 
single ones (and it should be noticed that from these the 
results of the ternary and higher interchanges may be 
obtained), and infers that 0 may be regarded as the root 
of a sextic of which the co-elficients are symmetric functions 
of the four arbitrary symbols. Mr. Cockle then shows that 
if we group the six forms of & two and two, the two members 
of each group being derivable one from the other by the 
conjugate interchanges, then the members of a group are 
^separable by any interchanges whatever that can be per- 
irmed upon the arbitrary symbols which enter into 6 . So 
hat symmetric functions of symmetric groups may be formed 
which are unsymmetric in 0, but yet unchanged by any 
permutations of the four arbitrary symbols. Consequently, 
if we apply the four arbitrary symbols as multipliers to four 
of the roots of a quintic, add the products to the fifth root and 
make the sum a constituent of the symmetric group-function 
will be a rational function of the fifth root, and therefore the 
root of a quintic into the co-efficients of which the arbitrary 
symbols enter symmetrically. In order to give the greatest 
simplicity to the sextic in 0 , the arbitrary symbols may have 
any suitable values assigned to them, and if we strive after a 
Symmetric Product we find that those values are unreal 
fifth roots of unity. Mr. Cockle adds, that the method of 
Symmetric Products has no special affinity for any particular 
theory of equations, and that although the evanescence of 
