655] 
A MEMOIR ON DIFFERENTIAL EQUATIONS. 
95 
as such, represented in the usual notation; thus if z = z(x,y), we have dz = ^dx + ^dy, 
where ^^ are the so-called partial differential coefficients of z in regard to x, y 
respectively. If we have y = y (x), then also dy = ^ dx, and the foregoing equation 
becomes 
, / dz dz dy\ 7 
dz = h- + - j- -N ) dx ; 
\dx dy dx) 
but considering the two equations z = z (x, y) and y = y(x) as determining z as a 
function of x, say z = z (x), we have dz = dx; whence comparing the two formulae 
d (z) _ dz dz dy 
dx dx dy dx ’ 
where is the so-called total differential. coefficient of z in regard to x. The 
d (z) dz 
distinction is best made, not by any difference of notation dx’ appending 
in any case of doubt the equations or equation used in the differentiation. Thus we 
have ~ where z=z{x, y)\ or, as the case may be, where z = z(x, y) and y — y (x). 
dx 
3. A relation between increments is always really a relation between differential 
coefficients: but we use the increments for symmetry and conciseness, as in the case 
of a differential system ~ ^ ^, or in a question relating to the lineo-differential 
Xdx + Ydy + Zdz, for instance in the question whether this can be put = du. 
Notations. 
4. Functional determinants. If a, 
then the determinants 
da 
da 
dx 
dy 
db 
db 
dx ’ 
dy 
are for shortness represented by 
d (a, b) 
d(x, yY 
Art. Nos. 4 to 6. 
i, c,... are functions of the variables x, y, z, w,..., 
da da da « 
dx’ dby’ dz ’ &C '’ 
db db db 
dx ’ dy ’ dz 
dc dc do 
dx’ dy’ dz 
d (a, b, c) , 
&0 -’