208 Proceedings of Royal Society of Edinburgh. [sess. 
been made be determines the sign of any particular term. On this 
point his words are (p. 202) 
“ Cela fait, determinons gdndralement le signe dn M me pro- 
duit (soit M) de la maniere snivante. Le nomhre M sera 
renferme entre les produits 1.2.3 ... I et 1.2.3 ...£(£+ 1); soit 
M=m + X x 1.2.3 . . . Z, de sorte que A<Z-fl, et m > 0 et 
<1 + 1.2.3 . . . 1. Cela etant, faisons M=m(- l)Vs (hi. 24) 
This apparently means that if the sign of the 23 rd term in the 
expansion of 
(abed, 1234)* 
be wanted, we divide 23 by 1.2.3, getting the quotient 3 and the 
remainder 5, and thence conclude that the sign wanted is got from 
the sign of the 5 th term by multiplying the latter by ( - l) 3 . Of 
course 5 has then to be dealt with after the manner of 23, the 
quotient and remainder this time being 2 and 1, so that we conclude 
that the sign of the 5 th term is got from the sign of the 1 st term 
by multiplying by (-1) 2 . And the sign of the 1 st term 
being + , the sign of the 23 rd is thus seen to be 
(-1) 3+2 le. 
It would seem at first as if the case where M is itself a factorial 
were neglected. This however, is not so, the condition m .< 1 + 
1.2.3 . . . I being corrective of the opening statement that M must 
lie between 1.2.3 . . . I and 1.2.3 . . . I (£+1). Lor example, the 
term being the 24 th , we put 24 in the form 3 x 1.2.3 + 6, and 
thus learn that the sign required is different from the sign of the 
6 th term : then we put 6 in the form 2 x 1.2 + 2, and thus learn 
that the sign of the 6 th term is the same as the sign of the 2 nd term ; 
finally, we put 2 in the form 1 x 1 + 1, which shows that the sign 
of the 2 nd term differs from the sign of the 1 st : the conclusion of 
the whole being that the signs of the 24 th and 1 st terms are the 
same, or that they are connected by the factor (-l) 3+2+1 . 
Though interesting in itself, a more troublesome form of the rule 
of signs for the purposes of demonstration it is scarcely possible 
to conceive, and, as might therefore be expected, it is on the score 
of logical development that Eeiss’ paper is weak. Through 
* Or ( abode ,12345) , or indeed (cqa 2 • • . a n) 123 . . . n). 
