210 
Proceedings of the Royal Society 
The proof may be easily given in a Cartesian form by operating by 
S i and S j separately. For the former operation gives 
d du d dv d d£ d dr) d 
dx ~ dx du + dx dv ~ dx dx dr)’ 
equations manifestly true. 
2. Now, the elementary area included by the curves u , u + Su, 
v , v + Sv, is easily seen to be 
8u8v 
TY Vu Vv ’ 
Hence we have the following transformations of a double integral 
extended over a given area : — 
a -jfi -Jfi tw w, ■ 
But by (1) we see at once that 
TVVf V^ = 
d£ d£ 
du * dv 
drj dr) 
du ’ dv 
TY VwVv , 
whence, of course, the general proposition 
d£ di 
du dv 
du ' dv 
M \ M 
dr) dr) 
du ’ dv 
1 
du dv 
dr) ’ dr) 
and the common transformation 
ffidxdy =fj P 
dx dx 
du ’ dv 
dy dy 
du ’ dv 
dudv . 
3. Dealing with triple integrals, V takes the ordinary Hamil- 
tonian form, and an additional term is added to each of the mem- 
bers of (1), which thus at once gives us the mode of introducing V 
into any system of curvilinear co-ordinates. 
