Prof. Sylvester on Derivation of Coexistence, 39 



^y. PD (« b c) indicates 



^2-fr • «l+;-"-^2-fr * ^1+,'+ ^2-j-r " ^2+r "~"^2+r * ^1-fr 

 + «2-j-r • ^1-l-r ~ ^2-}-r * ^l-|-r* 



I shall in general denote i^, PD {a b c ...I) actually eon- 

 panded as the zeta-ic product of ttfb^ c^ ... I in its rth phase. 



Art. (7.) General Properties of %eta-ic Products of Differences. 



If there be made one interchange in the order of the bases 

 to which ^ is prefixed, the zeta-ic product, in whatever phase 

 it be taken, remains unaltered in magnitude, but changes its 

 sign. 



Art. (8.) If in any phase of a zeta-ic product two of the 

 bases be made to coincide, the expansion vanishes. 



Art. (9.) Letyj be used, agreeably to the ordinary notation, 

 to denote the sum of the quantities to which it is prefixed, 

 y^ to denote the sum of the binary products, f^ of the ternary 

 ones, and so on. 



Thus lety*j (tfy bj Cj) or C (a b c) indicate a, + bi-{-Ci 



and r [Uf 5y f /) or f^ [a b c) indicate a^ bf + «y Cy + bj Cf 



and f («y b^ c^ or f^[ab c) indicate «y b^ c^ 



we shall be able now to state the following remarkable pro- 

 position connecting the several phases of certain the same 

 zeta-ic products. 



Art. (8.) Let a, b, c, ... /, denote any number of inde- 

 pendent bases, say (« — 1); but let the arguments of each base 

 be periodic, and the number of terms in each period the same 

 for every base, namely (»), so that 



a = a , = a 



r r-\-n r — n 



a = a 



n 



^ 



«-» 



b =b ^ =: b 



r r-\-n r—n 



b =b 



n 



= 



*-„ 



C = c . ■=. C 



r r-\-n r — n 



c — C 



n 



z:^ 



''-„ 



1=1^ = I 

 r r-\-n r — n 



I = I 



71 



— 



l-n 



r being any number whatever. Then 



^^iFD {oabc. .. I) =^(f^{abc... I) . ^VD {o abc.l)') 

 ^_2^T> {oabc...l)=^(^f^{abc...l) . ^FD {o a bc.l)) 



^-r'PT>{oabc..,l)==^(fr{abc„a) . ^FDioabc-l)} 



