405] 
AN EIGHTH MEMOIR ON QUANTICS. 
175 
by an equation of the form (u 2 , l) 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 , 1) 6 = 0; viz. one of these is 
u 12 
+ u 10 (a + 3 VA) 
+ u s (a — VA) 2 + A] 
— u s d 
+ m* [i (a + VA) 2 + A] A 
+ u 2 (a - 3 VA) A 2 
+ A 3 = 0, 
and the other is of course derived from it by reversing the sign of VA. 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, is 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 — - 8 s°- A), 
A = 5 *D, 
whence, in the notation of the present memoir, the expressions of these symbols are 
a = 5 4 J, 
d = - J5 10 (2 n L - J 3 - f JD), 
A = 5 5 D. 
305. From the equation in u, taking therein the radical VA as positive, M. Hermite 
obtains (d < 0 a mistake for d > 0) the following as the necessary and sufficient con 
ditions for the reality of all the roots, 
A = +, a + 3 VA = —, d = +, character or 
(Hermite’s third set of criteria). 
306. It is clear that a+3VA = - 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 or are 
D = +, J=-, 125J 2 — 9D = +, 2 n L - J 3 — fJi) = +. 
Now, writing as above, 
2»X - J 3 D 
x ~ J® ’ y ~J 2 ’ 
these are y = +, J = ~, H~~y = +, ^“1 V = -\ the conditions y = +, J=- imply that 
(x, y) is in the region P or the region Q; and the condition x — §y = — (observe the