154 
MR. A. C. DIXON ON SIMULTANEOUS 
Now in finding the value of the double integral taken over a part of any surface, 
it will be natural to suppose the co-ordinates of any point of such a surface to be 
functions of two parameters, say p, q, and to transform the integral into one taken 
with respect to these. The integral as transformed is 
and the known values in terms of p, q are to be substituted for x, y, z and their 
derivatives. 
The subject of integration is 
d(iv 
% 
, v) 
*) 
d(p,q) 
d± 
dO 
uyv) 0(£, 
x) 
> q) 
+ 
d(w 
d(x, 
’A 
v ) 
d(x, y) 
8( p, q y 
or 
dw 
dx 
dw dy 
+ 
dw 
dz 
dv 
dx 
+ 
dv 
3/y 
dv 
dz 
dv 
dp 
dy dp 
dz 
dp ’ 
dx 
dp 
dy 
dp 
dp 
d w 
dx 
dw dy 
+ 
d io 
8^ 
dv 
dx 
+ 
dv 
dy 
dv 
dz 
dx 
dq 
cly dy 
dz 
dq ’ 
dx 
dq 
dy 
dq 
+ 5 
dq 
d(w, v ) 
<Kp, q) ' 
The integral is therefore 
and if we take a single element we may write 
X dy dz + Y dz dx fi- Z dx dy — dw dv, 
dropping the parameters p , q, since the values which x , ?/, z have in terms of them 
are immaterial. 
This equation is meaningless unless the expression in terms of parameters is under¬ 
stood. The same is true of ordinary differentials. If when u is a function of x, y, z 
we write 
dn = fo dx 
ox 
, 7 , 
+ ck, d ' J + 
8 u 
dz 
dz 
we mean that if x, y, z 
parameter p, then 
are supposed to be any functions whatever of a single 
du du dx 
dp dx dp 
ck 
dp ' 
This equation being true quite independently of the expressions assumed for x, y, z in 
terms of p, we drop the denominator dp for convenience ; but in modern works on the 
Differential Calculus it is quite understood that a differential by itself is meaningless 
apart from this or some equivalent convention. 
