190 Proceedings of Royal Society of Edinburgh. [sess. 
notation for a genus of functions of which determinants, as under- 
stood up to the date of the paper, formed a species : thus 
a-J) 2 C 3 + «2^3 C l + a ^l C 2 ~ a ^2 C \ - Q 2pl C 3 ~ a A C 2 
is the case of 2 ±(123) where (123) = a-fige |. In the third 
place we are surprised to find that Cayley seems to propose to 
extend the meaning of the word determinant by transferring the 
name of the species to the genus, and to call by the name of “ ordi- 
nary determinants” the functions formerly known as “determin- 
ants ” merely. 
All this is in itself comparatively unimportant, serving perhaps 
only to recall to us Cauchy’s famous paper of 1812, where we have 
K, the originating term of an alternating function to compare and 
contrast with Cayley’s (12 . . . n), and ‘ alternating function ’ to com- 
pare and contrast with Cayley’s extended meaning of £ determinant.’ 
But what follows by way of second example is very noteworthy, 
because the originating term taken, viz., A 12 A 34 . . . \ n _ lin is one 
that could not possibly have been used by Cauchy, with whom 
2 denoted an operation of a much less simple character than per- 
mutation of the integers 1, 2, . . . , n. Unfortunately the example 
is not fully exploited.* We are only told that in a certain special 
* Supplying this defect we see that in strict accordance with Cayley’s 
definition 
12*34 
+ 
31*24 
- 12-43 
- 
31-42 
- 13-24 
- 
32-14 
+ 13-42 
+ 
32-41 
+ 14-23 
+ 
34-12 
- 14*32 
34-21 
- 21-34 
- 
41-23 
+ 21-43 
+ 
41-32 
+ 23*14 
+ 
42-13 
- 23-41 
- 
42-31 
- 24-13 
- 
43-12 
+ 24*31 
+ 
43 - 21 , 
2 { 12-34 
- 12-43 - 13-24 
+ 
13-42 
+ 14-23 
- 14-32 - 21-34 
+ 
21*43 
- 23-41 
+ 24*31 - 31-42 
+ 
32 - 41 }, 
— a function of twelve variables which is not a determinant in the acceptation 
either of the present time or of the time preceding Cayley. 
