1898-99.] Dr Muir on a Single Term of a Determinant. 473 
if we confine ourselves to effective interchanges we shall, in the 
course of the transformation of the permutation, meet with only 
two doubly-effective interchanges ; hut if, in selecting two elements 
of the second cycle, we make an ineffective interchange — say the 
interchange 6*^7, — the cycle is split up into 
and then we have three cycles of more than one element each, and 
therefore must be led later to three doubly-effective interchanges. 
(43) We have now dealt with four of the five rules of signs, and 
have seen how they are related to one another. Summing up, we 
may say that 8 being the number of “ inverted-pairs ” in a given 
permutation of n elements, v the smallest number of “ inter- 
changes ” necessary to transform the permutation into the standard 
permutation, k the number of “ circular substitutions ” needed for 
the same purpose, and g the smallest number of “ moves,” then 
Consequently, so far as we know, two of the four numbers are 
unrelated, namely, 8 and k. Of the former — the number of “ in- 
verted-pairs ” — a full investigation has been given above. In the 
case of the latter — the number of “ circular substitutions ” — such is 
unnecessary, as may be seen from several papers by Cauchy, the 
founder of the theory of substitutions.* The theorem of the fol- 
lowing § is the only fresh result which has been arrived at in the 
course of the . present investigation. 
(44) If the circular substitutions of any permutation be each 
reversed in order we obtain those of the conjugate permutation. 
In the substitution necessary to transform the given permutation 
into the standard permutation, any item, say A-into-/3, must have 
* See Muir, “History of Determinants,” pages 91 ... , 234 . . . , 259 . . . , 
or the papers themselves there referred to. 
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