38 Prof. Sylvester on Derivation of Coexistence. 



indices into their original position ; the result is the zeta-ic 

 product*." 



Thus for example ^(a^ . h^ is the same as simply a .b -, 



but ^ (a. . a^ represents not a^. a^ but a , . 



So in like manner 



r(K-i,)K-U) 



=^ a. . ,— a, .b — b, . a, •\- b . , 



r {{a, - bi) («/ - c^) {b, - c,)) 



= the depressed p?'odtict of (a — b) {a—c) {b—c) 

 — the depressed value of a?' {b' — c') + 6^ (c' — a') + c^ (a' — b') 

 i.e. = ttg .b^ — ac^. c^-^-b^, c^ — b^ . fti + Cg . a^ — Cc^.b-^. 



Art. (4.) We shall have occasion in this part to combine 

 the two symbols ^, PD : thus we shall use 

 ^ PD [a^ bl) to denote ^ {b^ — a/) 

 ^ PD («y bj €,) to denote ^ {{b^ — a^ (^/ — ^y) (^/ ~" ^i))* 



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 isfidly 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 t,. Thus 



^{ai—bi) [a^ — c) being equal to f/2 — «i . Cj + ^i . ^1 — ^1 . «i 

 f+i {a^ — bi) {a~c^ is equal to CTg— «2 . C2 4-^2 • ^2~"^2 • ^2 

 ^_i («/ — i/) {a, — c,) is equal to a^ — aQ . CQ-{-bQ. CQ—bQ.aQ, 

 In like manner ^ PD {a b c) indicating 



bc>. a^ — bc,. c^-^c^.bj—CQ . aj + ffg. Cj — «2 • ^1 



* It is scarcely necessarj' to add that an analogous interpretation may 

 be extended to any zeta-ic function whatever. Thus 



^ («j -1- ^1)2 = «2 + 2 «j *i 4- ^2 



