1898 - 99 .] Dr Muir on a Single Term of a Determinant. 469 
(36) From an examination of this process it will be seen that 
(a) A circular substitution is one in which the upper line of 
elements is got from the lower by removing the first 
element of the latter to the last place , and that conse- 
quently it may be represented by only one line , e.g ., 
1 
(b) There is only one way of decomposing a substitution into 
circular substitutions 
— a statement which includes the fact that 
(c) The number of circular substitutions necessary to transform 
any permutation into the standard permutation is 
definite. 
Further, it may be noted that a “ two-termed ” or “ binomial 
substitution, like ^ ^ is exactly the same as an “interchange 
11 
11 
or “transposition”; that a “ one-termed ” or “ monomial ” substi- 
tution is, strictly speaking, not a substitution at all : and that a 
circular substitution may he expressed in the two-line notation in 
as many ways as there are elements in each line. 
(37) We are now prepared for the important propositions which 
connect the number of “ interchanges ” necessary to transform a 
given permutation into the standard permutation with the number 
of “circular substitutions” required for the same purpose. The 
first is — • 
If in the lower line of a circular substitution of m elements two 
elements separated by p others be interchanged there is produced a 
substitution which can be resolved into two circular substitutions 
of m - (p + 1) and p + 1 elements respectively. 
Let the given circular substitution be 
TfryA*, , <£,x,tA,w,a\ 
\a,P,y,8,c, , 
