38 Prof. Sylvester on Derivation of Coexistence, 
indices into their original position ; the result is the zeta-ic 
product*.” 
Thus for example f is the same as simply 
but f (« . . represents not af but 
So in like manner 
? ({a, - b,) {a, - bj) 
= %+i ~^h-K- h • + K+k 
? ((«/ - ^() (“/ - c'!i {^; - c!)) 
= the depressed jproduct oi {a — h) {a—c) (b—c) 
= the depressed value of [If — cf) + [d — «') + [a! — V) 
l.e. Gfg • ^ j (X(^ , Cj “f" ^2 • h (2 • ~f" Cg • Ct-^ Cg * • 
Art. (4.) We shall have occasion in this part to combine 
the two symbols PD : thus we shall use 
f PD [a I to denote f (bj—a^ 
f PD [dj Cl) to denote f ((6y — [Cf — dl) [Cj — bf. 
Art. (5.) For the sake of elegance of diction I shall in future 
sometimes omit to insert the inferior index when it is unity ; 
but the reader must always bear in mind that it is to be under^ 
stood though not expressed. 
I shall thus be able to speak of the zeta-ic product of such 
and such bases mentioned by name. 
Art. (6.) We are not yet come to the limit of the powers 
of our notation. The zeta-ic product of the sum of arguments 
will consist of the sum of products of arguments, each argu- 
ment being (as I have defined) made up of a base and an infe- 
rior index. Now we may imagine each index of every term 
of the zeta-ic product after it is fully expanded to be increased 
or diminished by unity, or each at the same time to be in- 
creased or diminished by 2, or each in general to be increased 
or diminished by r. I shall denote this alteration by affixing 
an (r) with the positive or negative sign to the Thus 
i^[a—b^ [di — c,) being equal to ffg — ccj . Cj + Z>i . — 
f+i (^y — ^/) is equal to . C 2~ ^2 • ^2 
[d, — bi) {d, — c,) is equal to a^ — dQ . + . Cq— . cr^. 
In like manner f PD [d b c) indicating 
Z>2 . d^ — b^ . q -l-Cg . bj —c^ . «i- 4-«2 • ^i~^2 • ^1 
* It is scarcely necessary to add that an analogous interpretation may 
be extended to any zeta-ic function whatever. Thus 
^ («1 + = «2 + S + ^2 
