PROFESSOR W. M. HICKS OH VORTEX MOTIOH. 
39 
Comparing with (2) it follows that 
dn 
or 
Ji - 
We may regard then Etp (2) as replaced by (4), which includes it as the greater does 
the less. 
For the circuit for co sin y take a small circuit formed by a small arc ds of a meridian 
PP' on xp, the normals (drt,) at P. P' and the portion of the meridian arc on xp + dxp 
cut off by these normals. The flow along the normals dn is zero. Along ds it is 
V cos (p ds ; along ds' it is 
V cos (p ds 4- (v cos (p ds) dn. 
dn 
^ dn 
The area of the cross-section of ojsiiiy is dn ds. 
Hence 
But by (L), 
therefore 
d 
2oj sin y dn ds = — (r cos (p ds) dn 
V cos (p = 
‘Iirp 
Ttto) sin y ds — 
d 
dn 
Since dxpjds = (J xp wull give any component of velocity in the meridian plane in 
the same way as the ordinary stream-function. 
4. It will often be foiuid advantageous to express i// in terms of curvilinear 
co-ordinates. Denote these by u, v. Displacements perpendicular to the u will be 
denoted by dm, and to v by dn, to be estimated positive in the directions in which 
u, V respectively increase. 
The differential equation satisfied by xp is found by expressing the circulation round 
a small area bounded by the curves u, u -)- du, v, v dr dv. Let wj ( = w sin y) denote 
the rotation at a point of the area. We shall regard this as positive when it goes 
clockwise. The circulation is then 2w[ X area = 2(y, dn, dn. 
The velocities along PQ, PP' (see fig. 2) are respectively 
1 1 d'yp' 
'lirp dn 'lirp dn 
The riows along them are therefore (clockwise) 
1 
'lirp 
dll and + 
dn 2iTp 
dip 
dn' 
dn. 
