EXPRESSED IN TERMS OF THE ROOTS OF (^X, fx. 
435 
to denote the product of the differences resulting from the subtraction of each of the 
quantities X, in the lower line from all of those in the upper line I, m, w, ... p, 
the two values above given for may be written under the respective forms 
and 
^2 _ 
{X— hg) hg) ...{X- hg) 
X(x—n^){x—r;Q...{x-n^) 
in each of which equations disjunctively and in some order of relation each with each 
and 
^25 ^3 •••? 3 , ..., 171 , 
il5 ^25 ^3 •••5 ?» 1> '7) •••? 
These two forms are only the two extremities of a scale of forms all equally well 
adapted to express ; for let v and u be any two integers so taken as to satisfy the 
equation 
V + l> = i, 
and let R( ; ), where the dots denote any quantities whatever, be used to 
denote a rational form of function which remains unaltered in value when any two 
of the quantities under each and either (the same one) of the two bars are mutually 
interchanged, then we may write 
L 
kg hg ...hg 
^0 + 1 ^b + 2 
5./+1 iv + 3 
■ • (19-) 
For if, as above, we suppose x=hg=nc, any term of in which q^, ... comprise 
among them h^, or in which comprise among them riu, will vanish by virtue of 
the factors (x—hg){x—hg^)...{x—hgJx{x—7i^^{x — ri^^...{x—>^^); but if neither h^ nor 
riu is so comprised, then h^ must be one of the terms in the complementary seiies 
9 b+ 2 j and riu one of the terms in the complementary series | E/+1? ^i>' + 25 ••• 
and therefore one of the quantities hg^^^, ... hg^ will equal one of the quantities 
+i’ + 2 ’ consequently the term of 9^, in question will vanish by virtue of 
the factor 
hg hg ...hg 
1v + \ 1v + 2 
L |p+l + 2 In J 
vanishing. In either case therefore every term included 
within the sign of summation vanishes when x—hg — i. e. whenever y’(<r) 0 and 
p=i^x)=0. Hence as given by equation (19.), will satisfy the syzygetic equation 
