461 
1887.] Dr T. Muir on the Theory of Determinants. 
and it is pointed out that the first sign is always + , and the last 
+ or - according as the number 1 + 2 + 3+ . . . + (?z - 1) is even 
or odd. 
Bearing in mind that Hindenburg wrote his permutations in a 
definite order, this remark regarding the sequence of signs entitles 
us to view him as the author of a combined rule of term-formation 
and rule of signs, which may he formulated as follows: — 
Write the permutations of 1, 2, 3, . . ., n in ascending order of 
magnitude as if they were numbers; make the first sign + , the 
second — , the next pair contrary in sign to the first pair , the th ird 
pair contrary in sign to the second pair, the next six (1.2.3) contrary 
in sign to the first six , the third six contrary in sign to the second six, 
the fourth six contrary in sign to the third six, the next twenty - 
four (1. 2.3.4) contrary in sign to the first twenty-four , and so 
on. (n. 4 + hi. 5.) 
ROTHE, H. A. (1800). 
[Ueber Permutationen, in Beziehung auf die Stellen ihrer Elemente. 
Anwendung der daraus abgeleiteten Satze auf das Eliminations- 
problem. Sammlung combinatorisch-analytischer Abhand- 
lungen, herausg. v. C. F. Hindenburg, ii. pp. 263-305.] 
Rothe was a follower of Hindenburg, knew Hindenburg’s preface 
to Rudiger’s Specimen Analyticum, and was familiar with what had 
been done by Cramer and Bezout (see his words at p. 305). His 
memoir is very explicit and formal, proposition following definition, 
and corollary following proposition, in the most methodical manner. 
The idea which is made the basis of it, that of place-index 
(“ Stellenexponent ”), is an ill-advised and purposeless modification 
of Cramer’s idea of a “ derangement.” The definition is as follows: 
— In any permutation of the first n integers, the place-index of any 
integer is got by counting the integer itself, and all the elements after 
it which are less than it. For example, in the permutation 
6, 4, 3, 9, 8, 10, 1, 7, 2, 5 
of the first ten integers, the place-index of 9 is 6, and that of 7 is 3. 
The counting of the integer itself makes the place-index always one 
more than the number of “ derangements ” connected with the 
