1903-4.] 
Dr Muir on General Determinants. 
83 
and reducing the result by the equation (0) and the equations 
(6) , the second side becomes 
JC 0 0 . . . 
0 fl {r) v {r) . . . 
0 n {r+1) v {r+1 \ . . 
which is equivalent to 
,{r) 
,(r+l) 
,(r+ 1) 
or we have the equation 
A L 
* 
II 
fX {r) v (r) . . . 
1) v (r+1) . . . 
A (r-1) L (r - 1) 

which in the particular case of r = n becomes 
A .... B 
A' . . . . B' 
The Second Section is said to concern “the notation and pro- 
perties of certain functions resolvable into a series of determinants,” 
and it is at once seen that the functions in question are obtainable 
from the use of m sets of n indices in the way in which a deter- 
minant is obtainable from only two sets. Sylvester spoke of them 
later (1851) as commutants .* 
Cayley (1845). 
[On the theory of linear transformations. Carnb. Math. 
Journ ., iv. pp. 193-209 ; or Collected Math . Papers , i. 
pp. 80-94.] 
* See Postscript to Cayley’s paper “ On the Theory of Permutants,” Camb. 
and Dub. Math. Journ., vii. pp. 40-51 ; or Collected Math. Papers, ii. 
pp. 16-26. 
