1899 - 1900 .] Dr Muir on the Theory of Alternants. 
Ill 
find a generalisation of his which had no individual instances in 
support of it. There thus remains the curious and interesting 
question as to what amount of truth there is in the theorem as 
enunciated, and whether an amendment of the enunciation would 
not give something not merely unexceptionable but of important 
value. 
In trying to pass from symmetric functions like 'Za, %ab, %abc^ 
. . . which are linear in regard to each of the variables, and to 
extend the theorem to any symmetric function, Sylvester probably 
thought — at least it would be quite natural for him to do so — of 
expressing the latter in terms of the former and then applying the 
theorem already obtained. It is desirable, therefore, to see what 
such a process may lead to. Taking the case of the multiplier 
%a 2 bc we have 
i aWd 4 | . %a 2 bc = | aWd 4 1 . {%a . %abc - 4 %abcd} , 
= {| a}b 2 c*<T\.%a}.%abc - \aWcH\\:%abcd , 
= |aW<?J . %abc - 4|aWd 5 | . 
At this point we encounter a difficulty, for the previous theorem, 
although it teaches us to multiply |a 1 6 2 c 3 d 4 | by 2<ab, does not help 
us in the case where the multiplicand is |a 1 6 2 c 3 d 5 |. Proceeding, 
however, with other assistance we find the product 
= \a%^T\ + |aWd»| - 4|aWtf 6 [, 
= | aW^I - 3|aWd 5 | , . 
agreeing of course with what has already been found. Now the 
difficulty referred to would present itself to Sylvester also, but in 
a slightly different form by reason of the periodicity which he 
assumes in the elements. Thus, instead of writing 
{\a l b 2 c z d^\.%a}%abc = \a l b 2 c 3 d 5 |. %abc , 
= \aWd 5 \ + \amM Q \, 
he would write 
C{CPD(0a6crf).S 1 (a&crf)}.S 3 (afe^) = C{{-iPD(0aM).S 3 (aM)} 
and there pause for a little, not having specifically provided for the 
‘ zeta-ic ’ multiplication of such an expression as £_iPD(0rtM) by 
