1888 - 89 .] Dr T. Muir on the Theory of Determinants. 221 
occurs in connection with the subject of determinants, — a fact most 
significant of the comparative neglect of mathematical studies in 
Britain during the 18th century. Apart from the contents, there- 
fore, some little interest attaches to Drinkwater’s short paper, as 
being the first sign to us of that revival which, as is well known 
otherwise, had taken place some few years before. 
Drinkwater knew of the investigations of Cramer, Bezout, and 
Laplace; and professed only to put the elements of the subject 
“ in a more convenient form.” His rule of signs is stated and illus- 
trated as follows (p. 25) : — 
11 Write down the series of natural numbers 1 2 3 4. . . n, 
and underneath it all the permutations of these n numbers, 
prefixing to each a positive or negative sign according to the 
following condition : — ■ 
“ Any permutation may be derived from the first by con- 
sidering a requisite number of figures to move from left to 
right by a certain number of single steps or descents of a 
single place. If the whole number of such single steps neces- 
sary to derive any permutation from the first be even, that 
permutation has a positive sign prefixed to it ; the others are 
negative. For instance, 4 2 13. . . n may be derived from 
1 2 3 4. . . . n, by first causing the 3 to descend below the 4, 
requiring one single step : then the 2 below the new place of 
the 4, another single step ; lastly, the 1 below the new place 
of the 2, requiring two more steps, making in all 4. There- 
fore this permutation requires the positive sign.” 
In this there is essentially nothing new : it at once recalls a 
theorem of Rothe’s (in. 8). In the following paragraph, however, 
we find the discussion of a point not previously dealt with. The 
words are (p. 25) : — • 
“ The same permutation may be derived in various ways, and 
it is necessary, therefore, to show that this rule is not incon- 
sistent with itself : thus the same permutation 4 2 13. . . n 
might have been obtained by first marching 1 through 
three places, then 2 through two ; and, lastly, 3 through one, 
making six in all, an even number as before. Without accum- 
ulating instances, it is plain, if q be the smallest number of 
