694 
Proceedings of the Poyal Society of Edinburgh. [Sess. 
In the last three paragraphs he draws attention to the existence of 
expressions which may be viewed as “ determinant simbolici,” his first kind 
being those in which symbols of differentiation take the place of elements ; 
e.g. the expression 
whose vanishing is the condition for the derivability of the equation 
vdx + q dy + mz = o 
from a single primitive,, is denoted by 
I P — R 
r cy 
— a notation which is even less satisfactory than that for which it is a 
contraction, namely, 
p ® p 
dx 
q| Q 
R - R 
OZ 
The other kind of expressions originated with Binet, who in 1812 gave the 
identities 
&' = 2a26 - Zab , 
Zab'c" != ZaZb^c + IZabc - ZaZbc - SbZca - %e%ab ; 
but in this case, though the close resemblance of the right-hand expressions 
to the developments of axisymmetric determinants is pointed out, no 
notation founded on the fact is suggested. 
As an appendix there is a note on permutations, explaining circular 
substitutions, interchanges ( alternazioni ), inversions of order ( rovesciamenti 
d’ordine), and their relations to one another. Cauchy’s sign-rule depending 
on the number of circular substitutions is replaced by a simpler rule, which 
requires the counting of only the even circular substitutions. Thus the 
permutation 3265417 being got from the standard permutation 1234567 by 
means of the circular substitutions 
( 316 ), ( 2 ), ( 54 ), ( 7 ), 
and only one of these being even, the sign of 3265417 is ( — Bellavitis’ 
enunciation is : “ II numero delle alternazioni , con cni una disposizione pub 
mitarsi in un’ altra e pari o dispari insieme col numero di tutte le 
