540 PROFESSOR CAYLEX’S EIGHTH MEMOIR ON QU ANTICS. 
whence, in the notation of the present memoir, the expressions of these symbols are 
a= 5 4 J, 
d=-i5 10 (2 11 L-J 3 -|JD), 
A= 5 5 D. 
305. From the equation in u, taking therein the radical \/ A as positive, M. Heemite 
obtains (d < 0 a mistake for d > 0) the following as the necessary and sufficient condi- 
tions for the reality of all the roots, 
A= + , a+3\/A=— , d=+> character 5 r 
(Heemite’s third set of criteria). 
306. It is clear that a+3\/ A= — is equivalent to (a= — and a 2 — 9A=+), and we 
have a 2 — 9A=5 5 (125J 2 — 9D), so that these conditions for the character 5 r are 
D=+, J=-, 125J 2 -9D=+, 2 n L— J 3 — f JD= + . 
Now, writing as above, 
these are y=-\-, J = — , y=-\-, y— — ; the conditions y= +, J=— imply 
that (, x , y) is in the region P or the region Q ; and the condition x — fy= — (observe 
the line x—%y= 0 lies between the lines x-\-y= 0, x— 2y=0,and so does not cut either 
the region P or the region Q) restricts (x, y) to the region P ; and for every point of P 
y is at most =1, and the condition -f- 5 -— y—-\- is of course satisfied. The condition, 
125J 2 — 9D = -}- 5 is thus wholly unnecessary, and omitting it, the conditions are 
D=+, J=— , 2 n L— J 3 — f JD=0, character 5r, 
which, — f being an admissible value of p, agrees with the result ante , No. 283. 
307. It may be remarked in passing that if 12345 is a function of the roots (#,, x 2 , 
x 3 , # 4 , x 5 ) of a quintic equation, which function is such that it remains unaltered by the 
cyclical permutation 12345 into 23451, and also by the reversal (12345 into 15432) of 
the order of the roots, so that the function has in fact the 12 values 
a, =12345, j3j=24135, 
a 2 =13425, /3 2 = 32145, 
a 3 =14235, ^=43125, 
a 4 =21435, j3 4 =13245, 
a 5 = 31245, /3 5 =14325, 
a 6 =41325, /3 6 =12435, 
then <p(a, (3) being any unsymmetrical function of (a, /3), the equation having for its roots 
the six values of <p(a, (3) (viz. <p(a n /3J, <p(a 2 , (3 2 ) . . . <p(a 6 , (3 6 )) can be expressed ratio- 
