PROFESSOR CAYLEY’S EIGHTH MEMOIR ON QITANTICS. 
535 
294. If D= — , the character is 3r+2«, but if D=+, then expressing the foregoing 
form as a sum of four squares affected with positive or negative coefficients, the character 
will be 5 r or 2 -f- 4«, according as the coefficients are all positive, or are two positive and 
two negative. Whence, if N denote as above, then for 
D=-f-, N= — , D!=+, B= — , character is 5 r, 
D=+, N=— ,BDj= + l 
and [• character is r-\-M ; 
D= + ,N= + J 
and further, the combination D=+, N= — , D,= — ,B=+ cannot arise (Hermite’s 
first set of criteria). 
295. Again, from the equivalent form 
^|D 1 (f-D« 2 -f2DMw-10Aw 2 )+BD(10Av 2 -6fe-w 2 +9Dw 2 )|, 
which, if a are the roots of the equation 90 2 — 1OA0+D = O, is 
BDF 
( t—Zwvf—u 
then by similar reasoning it is concluded that 
D=+, 25A 2 — 9D= + , A=— , N= — , character is 5 r, 
I) = -f, 25A 2 — 9D=-j-, A= — , N=+,| 
D= + , 25A 2 -9D=+, A= + , „ r+M. 
D= + , 25A 2 — 9D = — , 
(Hermite’s second set of criteria). 
Article Nos. 296 to 303 . — Comparison with the Criteria No. 283: the Nodal Cubic. 
296. For the discussion of Hermite’s results, it is to be observed that in the notation 
of the present Memoir we have 
A =- J, 
B=— K=- t !s<J 2 — D), 
I) = 1 ). 
D,= 16L— JK= x ^ g -(2 11 L— J 3 + JD), 
N = 18L 2 — JKL— K 3 
= ^ ) 3 2 . 2 22 L 2 - 1 4 JL( J 2 — D)— (J 2 — D) 3 [. 
or, putting as above, 
2 n L — J 3 
J 3 
» and 1 +a— v , l —y— 
J 2 — D 
