448 Transactions of the Boyal Society of South Africa. 
the operations involved in (f){x)\p(y) - (p{y)\P{x) then performed, and the 
terms of the result combined in groups of four so as to admit of the factor 
y — x being struck out preparatory to making the substitution cc = i/ = a^. 
A much simpler mode v^ould have been to take (jj(x)\p'(x) - (p' (x)\p{x) as 
the limiting form of {(p[x)-^{y) - (j){y)-^{x)] [y - x), and then substitute 
for 0'(a,.), i^'(a,.) the expressions obtained from the interpolational forms of 
(l>{x), \ly{x) by dividing by x — a^ and thereafter putting cc = a^. 
3. As is often the case, however, the v^hole matter suffers as regards 
simplicity by reason of excessive specialisation. The follov^ing much more 
general theorem is susceptible of a much simpler proof. If ^(x, y) he used 
to stand for 
1 
a,. 
a^2. 
^271 
^n2 
and c^- for the dijference-p'oduct of any quantities following it, then 
l^a^a^,.. . ^{cioa^ — ^'{(loa, ... a„) . $(aoai) + i;^{aoa^a^ ... a^)^ (ttott^) 
- + (-l)"^K«oai ... a.-:) . ^KaJ = 0. 
By way of proof we have only to seek for the co-factor of a^^ on the left- 
hand side. Now in (^(x, y) this co-factor is seen to be y^~^x^~\ therefore 
the full co-factor sought is 
l,^{a^a^ ... . al 'al \ — 'C^a^a^ ... a^ . al 'a* ' + ^^(aoOittg 
- + ( - lfl,^{a^a^ . . . a„_,) . al-'a\ 
an) . ao 'a^ ' 
S — I 
which is seen to be the development of 
ar(-ir 1 
1 
al .. 
«! 
a] .. 
. ar' 
al . . 
. a'^~' 
arranged according to the elements of the last column of the determinant. 
As, however, s is not greater than 7i, the said last column must be identical 
with a preceding column ; and thus the theorem is proved. 
Borchardt's special case of this is where the square array in the 
expression of ^l^{x,y) is the peculiar axisymmetric array of Bezout's con- 
densed eliminant, for then by Cayley's theorem ^{x,y) becomes 
\ct>(x)xp{y)-(t>{y)xl.{x)\ -^(y-x) or ¥{x,y). 
