1887.] Dr T. Muir on the Theory of Determinants. 483 
sign of a in accordance with the first rule is the same as to count 
the number of letters placed under the given permutation, thus, 
b/Say 
aa . . 
P 
7 
in accordance with the second rule. 
Another point of resemblance between Rothe and De Prasse is 
thus made manifest, viz., that they both refused to accept Cramer’s 
rule of signs as fundamental, preferring to base their work on a 
rule equally arbitrary, and then to deduce Cramer’s from it. 
In case it may have escaped the reader, attention may likewise be 
drawn to the fact that De Prasse prefixes a sign not only to per- 
mutations involving all the letters dealt with, but also to any 
permutation whatever involving a less number ; so that in reckoning 
the sign of aSj3 , say, the full number of letters from which a, 8, /3 
are chosen must be known. 
A theorem like Hindenburg’s is next given, viz., If the permuta- 
tions of any group he separated into sub-groups, (1) those which begin 
with a, (2) those which begin with /3, and so on, then the series of 
signs of the 3rd, 5th, and other odd sub-groups is identical with the 
series of signs of the sub-group, and the signs of any one of the 
even sub-groups is got by changing each sign of the first sub-group 
into the opposite sign. (iii. 16.) 
It is more extensive than Hindenburg’s in that it is true of per- 
mutations which involve less than all the letters, provided such per- 
mutations have had their signs fixed in accordance with De Prasse’s 
rule. The proof depends, of course, on the first rule of signs, and 
consists in showing that if the theorem be true for any group it 
must, by the said rule, be true for the next group. It will be 
remembered that Hindenburg gave no proof. 
Following this is Rothe’s theorem regarding the interchange of 
two elements of a permutation, or rather an extension of the theorem 
to signed permutations involving less than the whole number of 
letters. The proof is as lengthy as Rothe’s, even more unnecessary 
letters than Rothe’s c, f, e being introduced. (hi. 17.) 
The last theorem is Vandermonde’s (xil); and this is followed by 
