462 Proceedings of Royal Society of Edinburgh. [july 18, 
integer. This necessitates the introduction of a corresponding 
modification of Cramer’s “rule of signs,” viz. 
“ 3. Willkiihrlicher Satz. Jede Permutation der Elemente 
1, 2, 3, . . . , r, werde mit dem Zeichen + versehen, wenn 
entweder gar keine, oder eine gerade Menge gerader Zahlen, 
unter ihren Stellenexponenten vorkommt ; mit dem Zeichen - 
hingegen, wenn die Menge der geraden Zahlen, unter den 
Stellenexponenten ungerade ist.” (hi. 6.) 
It is difficult to suggest any justification for the changes here intro- 
duced. The author himself refers to none. Indeed, in the very 
next paragraph he points out that to ascertain whether there be an 
even number of even integers among the place-indices is the same as 
to diminish each of the place-indices by 1, and ascertain whether 
there be an even number of odd integers, that is, whether the sum 
of the odd integers be even. He then concludes — 
“ Man kann also auch die Regel so ausdriicken: Jede Per- 
mutation bekommt das Zeichen + wenn die Summe der um 1 
verminderten Stellenexponenten gerade, — hingegen, wenn 
sie ungerade ist.” 
This is simply Cramer’s rule, and it is the only rule of signs 
employed henceforward in the memoir, the expression “ die Summe 
der um 1 verminderten Stellenexponenten,” occurring over and over 
again as a periphrasis for “ the number of derangements.” 
The next four pages are occupied with a very lengthy but 
thorough investigation of the theorem that two permutations differ 
in sign , if they be so related that either is got from the other by the 
interchange of two of the elements of the latter. Strictly speaking, 
however, the proposition proved is something more definite than 
this, viz,— 
If in a permutation of the integers 1.2, ... r there be d integers 
intermediate in place and value between any two , A and B, of the 
integers , the interchanging of the said two icould increase or dimmish 
the number of inversions of order by 2J + 1. (hi. 7.) 
The proof consists in finding the sum of the place-indices for the 
given permutation in terms of d as just defined, c the number of 
elements less than both A and B and situated between them, / the 
number of such elements situated to the right of B, and e the 
