40 
PROFESSOTl W. M. ETICRS ON VORTEX MOTION. 
Fig. 
2 . 
Heuce the total How round PQQ'P' is 
or 
rl / I d^lr j , 
— I r— a n 
an \l'np an 
du — 
dv 
/ 1 d\jr 
Tbi' 
d / 1 d^ du dn' 
du ylirp du dn dv 
du dv 
d / 1 d.-\^ 
dv \'27rp dv 
d:V 
ddd' 
But this is 2wi dn dn^. Hence 
d /I d->\r du dn'\^ _i_ ^ 
dv, \ p dv, ’ dn * dv j dv \ p dv dn'" du 
— dTTCUi 
dn dn' 
du dv 
— 47r&i sin ^ 
dn dn' 
du dv 
(5). 
In many cases p zl = f 'Vi-), giving duldn = dvldn, and the equation 
simplifies to 
y a m + 4 ,^, (*v 
du \ p du j dv \ p dv j \du / 
The following cases will he required :— 
(1) Cylindrical co-ordinates, (p, z), 
and 
du = dp = dn 
d 
1 dy{r\ 1 dV 
dp \ p dp j p dz- 
d"T^ 1 d'Kj/' d’yp' 
dp^ p dp dz- 
dv = dz = dn', 
= — Attoj sin y, 
= — 4:TTpoj sin y 
or 
( 6 ). 
