380 
PROFESSOR M. J. M. HILL OH THE MOTION OF FLUID, PART 
Observing that 
This may be written 
d d d \ 1 r 
dt +T dr +W Iz)r = --° 
r~ 
i +T i +a m( d ”^ 
dt' dr' dz)\r\dr dz 
d , d d\ 
dt dr 
therefore 
1 / dw dr 
r\dr dz 
(d , d , d\\ 
f dw diO 
\dt +T dr +W dz]\ 
dr dz) 
L r p J 
This becomes, on putting in Clebsch’s forms, t=^+X—^ y, 
fd d d \ f 1 (d\ d\jr d^jr dX\ I 
\dt 7 dr W dz)\rp\dr dz dr dz) J 
And as in Art. 2 this result may be deduced from the equation of continuity and 
, . , . dx dX dX . d-dr d-dr d-dr . 
the two equations H 7 + T-+W—= 0, and wt+t— +w-t~=0 only. 
dt dr d: 
8. Hence, supposing p constant 
X r ^jr z —X z ^fr r 
=f(K *)= 
Kg : -kg, = r 
b\}r 
Let differentials of the variables X, r, t, when regarded as independent be denoted 
by bX, dr, bt respectively, then 
f rbr bF(X, t ) 
i— ~ “ O • 
J (X> 
g-- 
bx 
and 
P /v „ A , bY(X,t) 
■ G(X, r, 0 + -^ 
1 r bG . b 2 F(X, t) 
T= ~ X 
bt 
X< 1 V 
w= — -— x, 
X- r 
btbX 
bG b 2 F(X, t) 
tf"*" btbX 
'To find the current function A there are the equations 
