1888 .] Dr Muir on a Class of Simple Alternants. 
307 
where, as usual, we put (a, h, cY for the complete symmetric function 
of a, &, c of the 4*^ degree. 
(8) When the symmetric function of all the variables hut one is 
fractional, difficulties arise. In many cases, it is true, a simple 
preparatory transformation does away with the fractional functions ; 
but even then considerable labour is necessary to reach a result 
which one feels certain must be obtainable in some much simpler 
way. As an example, let us take the determinant 
{ah + ac + hey 
“ ^ ^ {a + h)(a + c){h + c) 
chosen because it is sufficiently complicated, and because, thanks to 
a paper of Professor Anglin’s, the result is verifiable. Since 
(ab + ac + hcY — %a%^c^ + 31(a%-c + GaHi^c^ 
and (a + h)(a + c){b + e) = + ‘labc , 
it follows that 
{ab + + bcY %a%^c^^ 
{a + b)(a + c){b + c ) ''' + iahc ' 
The given alternant is thus equal to the sum of two alternants, viz. 
3 I c^ abc j , 
The second of these 
0 71 2 %a%^c^ 
TT7 ^77 ^ 
{a + b){a + c){b-{-c) 
— \a^ %a%^c^{d + a){d + b){d + c) | — 
where ’ = (a + b){a + c){a + d){b ^c){b + d){c + d) ; 
and since 
%a%h\d + a){d-vb){d-\-c) 
= %a%^c^{d^ + di^%aWc^ + d%abc^ + abc], 
= d^^a%^c^ + d\%a%H^ + %a%^ + d{%a^^c^ + ta^b^c) + %a%^c , 
the said second alternant 
= \aP b 1^2 d^%aWc^ + |a» dna^b^c^\\ 
+ |a0 c2 d^ta^^c I + |a0 d^a^^c^ 
+ \a^ c2 d%a%^c \ + |a^ b^ %a^b‘^c | j 
and this, as we learn from the above tables, 
