OF WHICH IS MOVING ROTATIONALLY AND PART IRROTATIONALLY. 379 
Then as before 
and 
therefore 
V 
Ah 
bGr 
bt 
= 0 
\ e SK(\, t) 1 cl ( r\ 
r dO 
© = rcb — A,. 
SK(A. t) 
SA 
= ~d)i K ^’ 
and the equation in A may be expressed in the form 
'' ! +;£+5£Xk(M-v»)=h(a,- 
v d?’ 3 1 r dr 1 r 3 d.6~)\ 
The current function is 
rbr AF(A, t ) 
J (Afljr ' ^>A 
K(A, t)--a 
To obtain this directly from the foregoing article, put for ~ its value — d> 
A® 
7. To obtain similar expressions when the motion of every element of the fluid is 
in planes passing through the axis of z, the motion being the same in all such planes. 
Let r be the velocity away from the axis of z, and w the velocity parallel axis of z. 
The equations of motion are 
'dp 
dr dr dr d 
~dt +T Tr + W di = ~Jr 
+V 
dw , dw d I [dp , , T 
+T *+"'* = -&(J7+ v 
dw . dw 
dt 
1 Id 
d 
dz 
r clr , dw 
A*+ t a. + “’-'-V+,:+,,.. + x-° 
r dr dz 
Differentiating the first equation with regard to 2, the second with regard to r and 
subtracting, it can be shown that 
therefore 
dr du A /dw dr\ ,/d d d\/dw dr\ 
Jr + & )\dr ~I4 + U +T A- +W &A* 
therefore 
dt 
+ 
d d\fdw dr\ fl fd , d , d\ t] (die 
T d r +W JdVr~^r\idt +7 dr +W i)P + r\[dr- 
dw dr 
dz 
1 fd d d\fdw dr 
die dr 
1 / dw dr 
d d d 
t _ _ j- . I- qjj - j Q 
r\dt ' ' dr 1 w dz)\dr dz) r~\dr dz) rp\dr dz)\\dt' dr' db>-'“ 
3 C 2 
