186 Proceedings of Royal Society of Edinburgh. [sess. 
0R 
0E 
0R 
= a 1 • + 
’ da lfS 
a *’da 2j + ■ ■ 
. . + C&2m,s » 
vcl 2m,s 
0R 
0E 
0R 
= + 
d^,: + ■ ■ 
+ 
a 
§ 
oj>I 
in the latter of which s differs from r, and the term a rs is 
oa rr 
awanting ; and finally, it is pointed out that the differential-quotient 
of R with respect to one or more elements are functions of the 
same kind as the original, and, probably as a consequence, that 
certain second differential- quotients are identical. No proofs are 
given ; indeed, the statements themselves are in the most concise 
form possible, the whole passage being as follows : — 
“ Designantibus i, i\ i'\ etc., indices inter se diversos, 
si sumuntur differentials partialia 
0R 8 2 R 
da iti , ’ da iti , ba i n i 5 
ea erunt aggregata ad instar aggregati R formata, respec- 
tive reiectis Coefficientium binis, quatuor, . . . seriebus 
cum horizontalibus turn verticalibus, eritque 
8 2 R = 8 2 R = 8 2 R . ” 
da iV ba u „ ba v „ Vt 0a M ,„ d&BL,, 
It should be carefully noted that both in this paper and in the 
preceding, Jacobi views the new functions as separate from and 
independent of determinants, and not at all in the light in which, 
at a later time, they came to be looked upon — viz., as a subsidiary 
function arising out of the study of a particular kind of determinant 
with which it had a definite quantitative relation. 
CAYLEY (1846). 
[Sur quelques proprietes des determinants gauches. Crelle’s 
Journ ., xxxii. pp. 119-123; or Collected Math. Papers , i. pp. 
332-336.] 
This paper, with its author’s usual directness, starts at once with 
a definition, the first words being — 
“Je donne le nom de determinant gauche a un deter- 
