526 
PROFESSOR CAYLEY’S EIGHTH MEMOIR ON QUANTICS. 
to 1)=+, these must have the characters 4 r and 4 i, and it is easy by considering a par- 
ticular case to show that B has the character 4 r, and A' the character 4 i ; C corresponds 
to D=— , and can therefore only have the character 2r+2«. Hence, for any given 
equation, (x, y, z) will lie in one of the regions (B, A', C), and if (x, y, z ) 
is in the region B , the character is 4 r, 
„ A', „ 4i, 
„ C, „ 2r+2i 
273. It is right to notice that the determination of the character is really made in 
what precedes ; the determination of the analytical criteria of the different characters is 
a mere corollary ; to obtain these it is only necessary to remark that 
2 = + , x=-\-, y=-\- includes the whole of facultative region B, 
that is, (x, y, z) being each positive, the character is 4r ; 
z=+, x=+, y= — 
x= 
x= 
{ include each a part and together the whole 
of facultative region A', 
that is, z being but (x, y) not each positive, the character is 4i\ 
z — — , x — +, y — -f- 
„ x= + ,y=~ 
I include each a part and together the whole 
1 of facultative region C, 
does not include any facultative space, 
that is, z being — , the character is 2r-\-2i ; and the combination of signs z= — , x= — , 
y=-\- is one which does not exist. 
The results thus agree with those furnished by Sturm’s theorem ; and in particular 
the impossibility of — , x~ — , y—-\- appears from Sturm’s theorem, inasmuch as 
his combination would give a gain instead of a loss of changes of sign. 
Article Nos. 274 to 285 . — Determination of the characters of the quintic equation. 
274. Passing now to the case of the quintic, I write 
J = No. 19, 
K= No. 25, 
1)= No. 26, 
L =— No. 29, 
I = No. 29A; 
viz. J is the quartinvariant, K and D are octinvariants (D the discriminant), L is 12-thic 
invariant, and I is the 18-thic or skew invariant. Hence also J, D, 2 n L— J 3 are inva- 
riants of the degrees 4, 8, 12 respectively; and forming the combinations 
2 n L— J 3 D 
