1904 - 5 .] Dr Muir on General Determinants. 939 
includes the case C) and the case C merit particular considera- 
tion. In fact, the case B is that of the functions which I 
have, in my memoir on Linear Transformations in the Journal , 
called hyperdeterminants, and the case C is the particular 
class of hyperdeterminants previously treated of by me in 
the Cambridge Philosophical Transactions , and also par- 
ticularly noticed in the memoir on Linear Transformations. 
The functions of the case B, I now propose to call ‘ Inter- 
mutants,’ and those in the case C, ‘Commutants.’ Corn- 
mutants include as a particular case ‘ Determinants,’ which 
term will be used in its ordinary signification.” 
To arrive at the position of determinants, therefore, in the great 
theory of permutants, we have first to seek out the particular 
permutants whose originating symbol is of the form 
p y ... . 
a ' P 7 
then in this restricted field to look for those in which the symbol 
just given is viewed as a product of symbols V a p y . , 
V a 'p y ....,• • • • ; next to confine ourselves in this smaller 
domain to those in which the ‘ blanks ’ of each single c set ’ 
form a separate column ; and lastly to isolate those in which 
the number of such columns is 2. 
In his paper Cayley goes on to expound the theory first of 
commutants , and then of intermutants. Neither exposition, how- 
ever, needs attention here, because the one has already been dealt 
with under the year 1843, and the other is outside our subject. 
Sylvester, J. J. (1852, Jan.). 
[On the principles of the calculus of forms. Cambridge and 
Dublin Math. Journ ., vii. pp. 52-97 ; Collected Math. 
Papers , i. pp. 284-327.] 
A postscript added by Cayley to his paper of the year 1851 
makes evident that Sylvester had a share in the latest generalisa- 
tion, and, as was natural, did not wish that share to be lost sight 
