52 
NOTE ON MR JERRARDS RESEARCHES ON 
[310 
and this is also the form of the other determinants, the only difference being as to the 
meaning of the symbol {a/3}, which, however, in each case denotes a function such that 
{a/3} = — {/3a}. Writing for greater shortness, 
{a/3 7 8e} = {a/3} + {/3 7 } + { 7 S} + {¿'e} + {ea}, 
and in like manner 
{a 7 e/3S} = {a 7 } + { 7 e} + {e/3} + {/38} + {¿a}, 
n, is an unsymmetric linear function (without constant term) of {a/3 7 8e}, {aye/38} ; or, 
what is all that is material, it is an unsymmetric function, containing only odd 
powers, of {a/3 7 8e}, {a 7 e/38}. 
If for 
a /3 7 8 e 
we substitute any one of the live arrangements 
a /3 7 8 e, 
/3 7 8 e a, 
y 8 e a /3, 
3 e a /3 ' 7 , 
e a /3 7 8, 
then {a/3 7 8e} and {aye/38} will in each case remain unaltered. 
But if we substitute any one of the five arrangements 
a e 8 7 /3, 
e 8 7 /3 a, 
8 7 /3 a e, 
7 /3 a e S, 
¡3 a e 8 y, 
then in each case {a/3 7 Se} and {oc 7 e/3S} will be changed into — {a/3 7 Se} and — {a 7 e/3S} 
respectively. Hence Hj remains unaltered by any one of the first five substitutions; 
and it is changed into — n x by any one of the second five substitutions. And the 
like being the case as regards n 2 , &c., it follows that the quotient n x -r-n 2 , or say P, 
remains unaltered by any one of the ten substitutions. Now the 120 permutations of 
a, ¡3, y, 8, e can be obtained as follows, viz. by forming the 12 different pentagons 
which can be formed with a, /3, y, 8, e (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.