522 
Proceedings of Boyal Society of Edinhurgh. 
/n 
77+1 
1\ 
^71 
77 + 1 
77+1 . 
f 71 
k 
k\ 
\a ; 
a , 
a) 
- a P( 
a\ 
a , 
a\ - . . 
a ; 
a , 
a) 
' n 
h 
iJ 
77+1 
^71 
h 
iJ 
77+1 
^77 
h 
id 
More than two pages are occupied with this part proof of Bezout’s 
recurrent law of formation. The identity of the terms of 
/W+l M + l\ 
Fi a ; s , a \ 
h h J 
with the terms of the numerator then follows at once; and the desired 
form for the value of a; , so far as the magnitude of the terms is 
concerned, is thus obtained. The corresponding forms for x^, . . ^ 
are of course immediately deducible. 
The rules for obtaining the terms of the numerator and 
denominator having been thus established in all their generality, 
the rule of signs is next dealt with. The treatment is cumbersome, 
but fresh and interesting. It is pointed out, to start with, that the 
counting of the inversions of order of a permutation, is equivalent 
to subtracting separately from each element all the elements which 
follow it, reckoning + 1 as a sign-factor when the difference is 
positive, and - 1 when the difference is negative, and then taking 
the product of all the said factors. This, it will be recalled, is 
essentially identical with an observation of Cauchy’s. Scherk, 
however, goes on to remark that these sign-factors may be viewed 
as functions of the differences which give rise to them, and may be 
so represented. Whether there actually be a function which 
equals q- 1 for all positive values of the argument and equals - 1 
for all negative values is left for future consideration. Cramer’s 
rule of signs is thus made to take the following form (p. 45) : — 
“ Wenn ^{P) eine solche Function der ganzen Zahl /5 isff 
welche= +1 ist, wenn j3 positiv, und- 1, wenn /? negativ ist, 
so ist das Vorzeichen Z irgend eines in dem Aggregate 
( 71 h h\ 
a\ a , a) enthaltenen Gliedes 
n k k ' 
123 k-\ k k-\ n 
a a a ... a a a ... a 
a a a'" aW aiW) a(") 
folgendes : 
