PROFESSOR CAYLEY’S EIGHTH MEMOIR ON QUANTICS. 
539 
3°. y— -f , — y= (x, y) lies beneath the line y= -f, viz. in one of the regions P, 
Q or T ; but J = -j- , (x, y) cannot lie in the region P or Q; hence the conditions give 
J=+, (x, y) within the portion which lies beneath the line y=Nf- of the region T. 
4°. y— +, — y = — , that is, (x, y) lies above the line y=-f, and therefore in one 
of the regions T or Q; and by the general theory, according as (x, y) lies in T or in Q, 
we shall have J=+ or J= — , hence the conditions give 
J = — , (x, y) within the portion which lies above the line y=^f, of the region Q. 
J=+, (x, y) within the portion which lies above the line y=^JL, of the region T. 
2°, 3°, and 4°, each of them agree with the character r+4i, and together they imply 
J= — , (x, y) any where in the region Q, or else J= +, (x, y) anywhere in the region T ; 
which is right. 
Article Nos. 304 to 307. — Hermite’s third set of Criteria; comparison with No. 283, 
and remarks. 
304. In the concluding portion of his memoir, M. Hermite obtains a third set of 
criteria for the character of a quintic equation ; this is found by means of the equation 
for the function 
of the roots (Q 0 , Q„ fl 2 , fl 3 , d 4 ) of the given quintic equation (a, b , c, d, e,ffQ, 1) S =Q. 
The function in question has 12 pairs of equal and opposite values, or it is determined 
by an equation of the form ( u 2 , 1) 12 =0, which equation is decomposable, not rationally 
but by the adjunction thereto of the square root of the discriminant, into two equations 
of the form ( u 2 , I) 6 =0 ; viz. one of these is 
u u 
+« ,# (a+3 N /A) 
+w 8 [i{a— A) 2 +A] 
- u { 5 d 
J r u ‘ i [i( a +\/ A) 2 +A]A 
4-w 2 (a— 3v^ A)A 2 
+ A 3 =0, 
and the other is of course derived from it by reversing the sign of A. I have in the 
equation written (a, d) instead of Hermite’s writing capitals A, D ; the sign — of the term 
in u 6 instead of +, as printed in his memoir, js a correction communicated to me by 
himself. The signification of the symbols is in the author’s notation 
a=5 4 A, 
d=4.5 9 (AD— ^-Dj), 
A=5T), 
4 d 2 
