728 :ME. a. CAYLEY OX THE COXDITIOXS FOE THE EXISTEXCE OF GITEX 
may be said to be complementary to each other. A particular product (a— j 3 )( 7 — 5) 
does or does not enter as a term (or factor of a term) in one of the above-mentioned 
sums, according as the type 12.34 of the product, or some similar type, does or does 
not form part of the type of the sum; for instance, the product (a — ^)(y — o) is a term 
(or factor of a term) of each of the sums 12.34, 13.45.24, &c., but not of the sums 
12.13.14.15, &c. 
2. If, now, we establish .any equalities between the roots, e. g. a-=(5, y=^, the effect 
will be to reduce certain of the sums to zero, and it is easy to ffnd in what cases this 
happens. The sum will vanish if each term contains one or both of the factors 
0,-13, y— ^. e. if there is no term the complementary of which contains the product 
(a — j3)(y — h), or what is the same thing, whenever the complementary’ type does not 
contain as part of it, a type such as 12 . 34. Thus for the sum 14 . 15 . 24 . 25 . 34 . 35 . 45. 
the complementary type is 12.13.23, which does not contain any type such as 12.34, 
e. the sum 14.15.24.25.34.35.45 vanishes for a=(3, y='h. It is of course clear 
that it also vanishes for ci=(3=s, 7 =^ or cc=f3 = y = ^, &c., which are included in a=/3, 
7 =^. But the like reasoning shows, and it is important to notice, that the sum in 
question does not vanish for c(=j3 = y: and of course it does not vanish for a=f3. 
Hence the vanishing of the sum 14.15.24.25.34.35.45 is characteristic of the system 
cx,=j3, 7 =^. A system of roots a, (3, 7 , i may be denoted by 11111 ; but if a=i3, then 
the system may be denoted by 2111 , or if c 4 =/ 3 , 7 =^, by 221 , and so on. AVe may then 
say that the sum 14.15.24.25.34.35.45 does not vanish for 2111, vanishes for 221. 
does not vanish for 311, vanishes for 32, 41, 5. 
3. For the purpose of obtaining the entire system of results it is only necessary to 
form Tables, such as the annexed Tables, the meaning of which is sirfhciently explained 
by what precedes : the mark ( x ) set against a type denotes that the sum represented 
by the complementary type vanishes, the mark (o) that the complementary tvpe does 
not vanish, for the system of roots denoted by the symbol at the top or bottom of the 
column ; the complementary type is given in the same horizontal line uith the original 
type. It will be noticed that the right-hand columns do not extend to the foot of the 
Table ; the reason of this of course is, to avoid a repetition of the same type. Some of 
the types at the foot of the Tables are complementary to themselves, but I have, not- 
withstanding this, given the complementary type in the form under which it naturallv 
presents itself. 
4. The Tables are — 
Table for the equal Hoots of a Quartic. 
211 
22 
31 
4 
211 
22 
31 
4 
X 
X 
X 
X 
12. 13.14 
.23.24.34 
0 
0 
0 
0 
0 
X 
X 
X 
12 jl3.14 
.23.24.34 
0 
0 
0 
X 
0 
X 
X 
X 
12.iT| 14 
.23.24.34 
0 
0 
0 
X 
0 
0 
X 
X 
12.34I 13 
. 14.23.24 
0 
0 
X 
X 
0 
X 
X 
X 
12.13.14 
23.24.34 
0 
0 
X 
X 
12.13.24 
14.23.34 
211 
22 
31 
4 
0 
X 
0 
X 
12.13.23 
14.24.34 
211 
22 
31 
4 
