466 
ON THE CONDITIONS FOR THE EXISTENCE OF GIVEN 
[150 
syzygetie relations between the different functions which vanish for any particular 
system of equalities, and of the order of the system composed of the several conditions 
for the particular system of equalities, does not enter into the plan of the present 
memoir. I have referred here to the indeterminateness of the question for the sake of 
the remark that I have availed myself thereof, to express by means of invariants or 
covariants the different systems of conditions obtained in the sequel of the memoir; the 
expressions of the different invariants and covariants referred to are given in my ‘Second 
Memoir upon Quantics,’ Philosophical Transactions, vol. cxlvi. (1856), [141]. 
1. Suppose, to fix the ideas, that the equation is one of the fifth order, and call 
the roots a, /3, 7, 8, e. Write 12 = 1cf> (a — ¡3) 1 , 12.13 = 1(f>(a — f3) l (a — >y) m , 12.34 = 
Icf) (a — /3) l (7 — 8) n , &c., where </> is an arbitrary function and l, m, &c. are positive integers. 
It is hardly necessary to remark that similar types, such as 12, 13, 45, &c., or as 12.13 
and 23.25, &c., denote identically the same sums. Two types, such as 12.13 and 
14.15.23.24.25.34.35.45, may be said to be complementary to each other. A par 
ticular product (a — ¡3) (7 — 8) 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 —/3) (7 — 3) 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 = /3, 7 = 8, the 
effect will be to reduce certain of the sums to zero, and it is easy to find in what 
cases this happens. The sum will, vanish if each term contains one or both of the factors 
a — /3, 7 — 8, i. e. if there is no term the complementary of which contains the product 
(a. — /3) (7 — 8), or what is the same thing, whenever the complementary type does not con 
tain 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, 
i. e. the sum 14.15.24.25.34.35.45 vanishes for a — /3, 7 = 8. It is of course clear 
that it also vanishes for a = /3 = e, 7 =8 or a = f3 = y = 8, &c., which are included in 
a = /3, 7 =8. But the like reasoning shows, and it is important to notice, that the 
sum in question does not vanish for a. = /3 = 7: and of course it does not vanish for 
a = 13. Hence the vanishing of the sum 14.15.24.25.34.35.45 is characteristic of the 
system a = /3, 7=S. A system of roots a, /3, 7, 8, e may be denoted by 11111; but 
if a = (3, then the system may be denoted by 2111, or if a = /3, 7 = 8, by 221, and 
so on. We 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 sufficiently 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 type 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 with 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