NOTE ON THE H YDRODYN AMICAL EQUATIONS. 
7 
890 
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 ' 
Representing for shortness the Matrix 
du du du 
dx ’ dy ’ dz 
dv dv dv 
dx dy ’ dz 
dw dw dw 
dx ’ dy ’ dz 
by 
a , b , c 
a', b', d 
and its square by 
A , B , G 
A', B', C 
we have 
a", b", c" 
A", 
B", 
G" 
ldu dv 
dw\ 
/du 
dv 
dw\ 
/du 
dv 
dw\ 
\dx ’ dx ’ 
<to)’ 
\dy ’ 
dy’ 
W’ 
\dz ’ 
dz ’ 
dz) 
A , B , 
C 
du 
du 
du 
A', B', 
C' 
dx ’ 
dy’ 
dz 
>1 >> 
A", B", 
G" 
dv 
dv 
dv 
dx’ 
dy’ 
dz 
dvj 
dw 
dw 
dx ’ 
dy’ 
dz 
” ” ” 
viz. the combinations which enter into the foregoing formula are 
and 
Q, 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 — b") + C' — B" = 0 ; viz. the three equations are 
D (c - b") + C' - B" = 0, 
D(a"-c ) + A" — C =0, 
D (b — a) + B — A' = 0, 
which are the equations in question. 
Observe that we have 
C' — B" = (a', b', c")(c, c\ c") — (a", b", c")(b, b', b") 
= a'c' + b'c + c'c" - a"b - b'b" - b"c", 
and thence, writing 
we have 
p — a(c' — b") + b (a" — c) + c (6 — a'), 
— ad — ab" + a"b — a'c, 
C'-B" + p = (a+b'+ c") (c - b") = 0,