39 
Prof. Sylvester on Derivation of Coexistence* 
PD (a b c) indicates 
bfi I • a* I ““ bf \ « • C| I ~f” 1 . b^ ■ *■“" i 
2-{-r 2+r 2-j-r 2+r 2+r 1 + ^ 
”1” I * C-t , ““ Ct\y 1 . I . 
* 2-f-r 1+r 2+r 1+r 
I shall in general denote PD {a b c ... /) actually ex~ 
panded as the zeta-ic product of a^b^c^ ... I in its rih. phase. 
Art. (7.) General Properties of Zeta-ic Products of Differences, 
If there be made one interchange in the order of the bases 
to which f is prefixed, the zeta-ic product, in whatever phase 
it be taken, remains unaltered in magnitude, but changes its 
sign. 
Art. (8.) If in ?Luy phase of a zeta-ic product two of the 
bases be made to coincide, the expansion vanishes. 
Art. (9.) Let be used, agreeably to the ordinary notation, 
to denote the sum of the quantities to which it is prefixed, 
to denote the sum of the binary products, of the ternary 
ones, and so on. 
Thus let /j {a, b, c,) or (« b c) indicate a^ + bj + c^ 
and («/ bi c^) or {a b c) indicate Uf b^ -\-a^c^-\- b^ c^ 
and («y bf c^) or f^[a b c) indicate b^ Cf 
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, ... I, 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 (n), so that 
a = a , —a 
r r-j-n r — n 
b = b ^ ^b 
r r-\-n r—n 
C = C . = C 
r r-\-n r — n 
C 
n 
C 
0 
I — I , z= / I — I =Z I 
r r-\-n r—n n o — n 
r being any number whatever. Then 
PD {pabc,,,l) = {ab c,,,l) . {o a b c,,.l)^ 
f _2 PD [o ab c,,,l) ^(^J^[ab c,„l) . [o a b c.„tj^ 
f-r PD {oab c,„l) = f (y^ {a b c,,d) . ? PD (pab c,,,l)) 
