212 Mr. A.Cayley’s Note on Mr. Jerrard’s Researches 
{aPryde}, Sauye8o' ; or, what is all that is material, it is an un- 
symmetric function, containing only odd powers, of {aByéde}, 
fayeBo', 
If for aByde 
we substitute any one of the five arrangements 
aBy de, 
Bydea, 
y dea, 
o ea 6 y, 
ea By 6, 
then {aByde} and {aye8o} will in each case remain unaltered. 
But if we substitute any one of the five arrangements 
aedy B, 
E4098 we, 
oy fae, 
yBae o, 
Baedy, 
then in each case {aPrydet and {ayeB8o} will be changed into 
— SaBydel and — }ayeBo} respectively. Hence I, remains un- 
altered by any one of the first five substitutions ; and it is changed 
into —II, by any one of the second five substitutions. And the 
like being the case as regards II,, &c., 1t follows that the quotient 
a or say P, remains unaltered by any one of the ten substitu- 
tions. Now the 120 permutations of «, 8, y, 6, « can be ob- 
tained as follows, viz. by forming the 12 different pentagons 
which can be formed with @, 8, y, 6, € (treated as five points), 
and reading each of them off in either direction from any angle. 
To each of the 12 pentagons there corresponds a distinct value 
of P, but such value is not altered by the different modes of 
reading off the pentagon ; P is consequently a 12-valued function. 
But there is a more simple form of the analytical expression 
of such a 12-valued function ; in fact, if [a@@yde] be any func- 
tion which is not altered by any one of the above ten substitu- 
tions—if, for instance, [#8] is a symmetrical function of x, @g, 
and 
[«Byde] = [a8] + [By] + [8] + [Se] + [ex], 
[eye8d] = [ay] + [ye] + [8] + [80] + [de], 
then any unsymmetrical function of [aSyde] and [ayeBd] will 
be a 12-valued function homotypical with P. 
an 
