6 
[890 
890 
where the t 
expression oi 
890. 
NOTE ON THE HYDRO DYNAMICAL EQUATIONS. 
[From the Proceedings of the Royal Society of Edinburgh, vol. xv. (1889), pp. 342—344.] 
Representing 
c 
c 
c 
c 
o 
c 
we have 
cl cl cL cl 
Writing for shortness = ^ + then if from the hydrodynamical 
equations 
Du = 
dx 
V —P 
P 
Dv = 
dy 
v --f Dw =i( v ~-P- 
without the aid of the equation 
du dv diu __ n 
dx + dy ' dz 
we eliminate V — -, we obtain equations not equivalent to those of Helmholtz, 
/ rf , d d\ du dv dw 
D %-\% dx + v dy + ^ dz) U ’ % dx + v dx + ^ dx’ & ’ 
^ dv dw dw du du dv , , , . , ,. ., , 
(2f, 2rj, —~j— , , ~j , as usual), but which, transforming them by 
dz dy dx dz dy dx 
means of the omitted equation, agree as they should do with his equations. But 
the form of the equations obtained directly by elimination as above, is an interesting 
one, which it is worth while to give. 
We have 
[dv dw 
\dz dy 
n [dv dw 
jdz dy 
-^Dv + ^Dw, 
dz dy 
d d 
: + “a 
d fdv 
d 
d 
-j: + U -j 1- V j b W , I I 7 
dt dx dy dzl \dz 
dv dw 
dy 
dv 
dv dv 
dz \dt + a dx^ V dy + W dz 
d [dw dw dw dw 
+ dy{-dt +U Tx +V dy +W Tz 
A , J 
A', J 
A", j 
viz. the com! 
and 
and the equ 
which are th< 
Observe 
and thence, v 
we have