1901 - 2 .] 
Dr Muir on the Theory of Jacobians. 
161 
v k = a k W + a k w 1 + a k w 2 + . . . + aj? l) lO n _ x 
ft 
du-, du c 
2±«- x ™ 2 
dv 2 
du^ 
dv„ 
Z± 
a af) . . . cLn-i^ 
dW»-l 
to which is added the remark that if there were one additional 
independent variable v we should similarly have 
du du x 
^ ~dv 0iq 
du n _ x 
dvff x 
dw dw 1 
~ dv dv x 
Sv’n-l) 
Sv n J 
Jacobi (1841). 
[De determinantibus functionalibus. Grelle’s Journ ., xxii. 
pp. 319-359.] 
Up to this point, as will have been evident, the special deter- 
minants which we are considering have turned up merely inciden- 
tally in the coarse of other work. Now, however, we come upon 
a separate and direct investigation of their properties, the memoir 
under consideration being the second of the three portions into 
which Jacobi divided his formal exposition of the theory of 
determinants. From the mere fact that separate treatment is be- 
stowed by him on only one other special form, it is clear that the 
subject of the memoir had come to be considered of particular 
importance. The same is rendered still more strikingly apparent 
when it is recalled that of the 87 pages occupied by the whole 
exposition, as many as 41 are devoted to this second portion con- 
cerning a subordinate form, while only 34 are assigned to what we 
are bound to consider the main portion, viz., that dealing with 
determinants in general. 
At the outset the preceding memoir 1 De formatione et pro- 
prietatibus determinantium ’ is referred to, and intimation made 
that there is now about to be considered the special case where the 
elements are partial differential-quotients of a set of n functions 
each of the same n independent variables, and that in this case 
the special name functional determinants may with convenience 
be used. Jacobi takes pains, however, to explain that this relation 
of general to particular may appropriately be taken in reverse 
order, going, in fact, so far as to say that from the properties of 
functional determinants the properties of what he calls algebraic 
PROC. ROY. SOC. EDIK — VOL. XXIV. 11 
