162 
Proceedings of Royal Society of Edinburgh. [sess. 
determinants may be deduced. He is careful to note also another 
relationship of the same kind, his statement being that in various 
questions relating to a system of functions the functional deter- 
minant is the analogue of the single differential-quotient in the 
case of a function of one variable. 
The subject of the notation of partial differential-quotients is 
then entered on at some length (pp. 320-323), and the decision 
made to use 0 in the manner which soon afterwards came to be 
familiar. The insufficiency of this notation is not forgotten, how- 
ever, although its advantages over the different devices of Euler 
and Lagrange are recognised, his illustrative example being the 
dz 
case of — where z is a function of x and u, and u is a function of 
dx 
x and y. He puts the whole matter in a nutshell when he says 
that it is not enough to specify the function to be operated on and 
the particular independent variable with respect to which the 
differentiation is to be performed, but that it is equally necessary 
to indicate the involved quantities which are to be viewed as con- 
stants during the operation.* 
* I may state in passing that in 1869 when lecturing on the subject I found 
it very useful to write 
4>x, y, z , Ts, t, u, v , .... 
in place of 
<l>m y, *) , /(s, t,u,v), . . . . 
and then indicate the number of times the function had to be differentiated 
with respect to any one of the variables by writing that number on the 
opposite side of the vinculum from the said variable ; thus 
1 2 3 
4>x,y,z 
meant the result of differentiating once with respect to x, thrice with respect 
to y , and twice with respect to z. 
Using this notation to illustrate Jacobi’s example, we see that if it were 
given that 
Z — 4>X , u 
we should have 
dz_ _ i 
dx ~ < r >x > u ’ 
but that if it were given that 
z=4 >x > u 
then we should not be certain as to 
and u = if/x,y 
dz 
the meaning of ^ , as it would stand for 
dx 
i l l l 
<f>x,u or <f> x > u + <t> x ,u * 'I'XiV 
according as u or y was to be considered constant. 
