1888.] Professor Cayley on Hydrodynamical Equations. 343 
d fdv 
d^dt 
dv dv dv 
w-T- + Oy- + 
ax ay dz 
) 
d ( dw dw dw dw\ 
Wy<dt'^'^dx “'■& ) ’ 
where the terms containing second derived functions disappear of 
themselves, and the expression on the right hand is thus 
du dv dv dv dw dv 
dz dx dz dy dz dz 
du dw dv dw dw dw 
dy dx dy dy dy dz ' 
Represent for shortness the Matrix 
du du du 
dx ^ dy ^ dz 
dv dv dv 
dx ^ dy ' dz 
dw dw dw 
dx ^ dy ^ dz 
a, h, c 
, and its square by 
A, B, C 
a', d 
A', B', C' 
a!\ h", c” 
A", B", C" 
we have 
A, B, C 
A', B', C' 
A", B", C'' 
( du dv dw\ fdu dv dw\ /du dv dw 
dx^ dx^ dx)^\dy^dy^dyj'\dz’dz’ dz . 
du dv dw'' 
dx^ dx^ dx^ 
du du du 
dx^ dy' dz 
dv dv dv 
dx' dy^ dz 
dw dw dw 
dx^ dy^ dz 
viz., the combinations which enter into the foregoing formula are 
p, _ dv du dv dv dv dw , 
dx dz ^ dy dz^ dz dz ' 
_ dw du dw dv dw dw 
dx dy dy dy dz dy ' 
and the equation thus is D(c' - 6") + C' - B" = 0; viz., the three 
equations are 
D(c' -h") + C -B" = 0, 
D(a"-c) +A"-C =0, 
D(c -a') +B -A' = 0, 
which are the equations in question. 
Observe that we have 
