174 
AN EIGHTH MEMOIR ON QUANTICS. 
[405 
303. Hermite’s second set of criteria are 
y—+, ^ — y=+, P=—, ^(x, y) = ~, character or, 
y=+> 2 £-y =+, J=~, ^(^2/)=+! 
y — -\-, 2/ = +, J = +, >■ character r + 4i. 
y = +, ¥-2/=-. J 
1°. If y = +, ^ (x, y) = ~, then the point (x, y) must be situate within the loop 
or within the triangle; and recollecting that at the highest point of the loop we have 
y = 2*. j the condition ^ — y=+ is satisfied for every such point, and may therefore be 
omitted. The conditions therefore are J=—, (x, y) within the loop, that is, in the 
region T, or within the triangle, that is, in the region P or the region T; but for 
any point of T the general theory gives J=+, and the conditions are therefore J=—, 
(x, y) within the region P; which agrees with the character or. 
2°. y =+, ^(x, y) = + , that is, (x, y) is within the upper region, that is, in the 
region Q or T] and — y=+, (x, y) will be within the portions of Q and T which 
lie beneath the line y = 2£; but J= — , and therefore (x, y) cannot lie in the region 
T\ hence the conditions amount to J=—, (x, y) within that portion which lies 
beneath the line y = ^ of the region Q. 
3°. y — +, 2 £- — y = +, (<¡0, y) lies beneath the line y = viz. in one of the 
regions P, Q or T; but J = +, (x, y) cannot lie in the region P or Q; hence the con 
ditions give J=+, (x, y) within the portion which lies beneath the line y = of the 
region T. 
4°. y = +, ^ — y = —, that is, (x, y) lies above the line y = 2 $-> and therefore in 
one of the regions T or Q; and by the general theory, according as (x, y) lies in T 
or in Q, we shall have J = + or J = -, hence the conditions give 
J——, (x, y) within the portion which lies above the line 2/ = ur> °f the region Q. 
J = +, (x, y) within the portion which lies above the line y = 2 £, of the region T. 
2°, 3°, and 4°, each of them agree with the character r + 4i, and together they imply 
J = —, (x, y) anywhere in the region Q, or else J=+, (x, y) anywhere in the region T; 
which is right. 
Article Nos. 304 to 307.—Hermite’s third set of Criteria; comparison with No. 283, 
and remarks. 
304. In the concluding portion of his memoir, M. Hermite obtains a third set of 
criteria for the character of a quintic equation ; this is found by means of the equation 
for the function 
a 4 (d 0 - 0 X ) (6, - 0 2 ) (d., - d 3 ) (d, - d 4 ) (d 4 - d 0 ) 
of the roots (d 0 , d l5 d 2 , d 3 , d 4 ) of the given quintic equation (a, b, c, d, e, /$d, l) 5 = 0. 
The function in question has 12 pairs of equal and opposite values, or it is determined