252 CONSTRUCTION OF VORTICES IN EARTH’S ATMOSPHERE 
u aw . v 
— = — tan i = — tan i 
w 2 w 
Hence, tan i 
1 2 Tttr u 
2 u 
= -, and a tan i = the 
a'tt'wZ'w aww 
ratio of the line integral to the surface integral. 
Similarly from (523) to (525) for the funnel-shaped vortex, 
and from (527) to (529) for the dumb-bell-shaped vortex, we 
may summarize, 
* » Lv 
(652) - 
Funnel Vortex 
1 
u 
V 
u 
w 
V 
w 
w = 
Dumb-bell Vortex 
u 
- = — cot az = tan i. 
w 
2’ 
ter 
2' 
2 
ter 
ter ter 
— — a cot az = — a tani. 
A A 
ter 
2 fl ' 
1 
— . z . u. 
w = - 
ter u tan z 
The connecting link between these vortices becomes, 
(653) — — = tan i, 
v 7 az 
the difference of sign depending upon the fact that the vertical 
axis was assumed in opposite directions in these vortices. The 
tangential angle i varies from one plane to another, — 90° on 
the lower reference plane, gradually changing to 0° on the middle 
plane where there is no inflowing or outflowing air, then con 
tinuing to + 90° on the upper reference plane. These are due 
entirely to the supply of air needed to balance the inflowing or 
outflowing line integrals with the increasing or decreasing vertical 
surface integrals over the same planes in succession. 
It has been customary for meteorologists to explain these 
tangential angles in cyclones as the effect of the deflecting or 
the friction forces. Thus, by equations (480) (481), 
/X . u 
(654) tan^i ; 
k — c 
—, for the inner part, 
u 
X 
k r , 
-j, for the outer part, 
but in fact the theory is erroneous. The deflecting forces de 
pendent upon X are small, and those depending on the friction 
tan ¿2 = - = 
v