360 SMITHSONIAN MISCELLANEOUS COLLECTIONS VOL. 5 1 



This equation shows us that for every given symmetric circular 

 distribution of pressure there is a system of velocities, for which the 

 whirl becomes a centered whirl, and that, conversely, for even* 

 system of uniform circular motions about one and the same axis, 

 provided these merge continuously into each other, there is a definite 

 distribution of pressure for which these motions are permanent or 

 steady, that is to say, they correspond to the conditions of the cen- 

 tered whirl : but of course this is true only in so far as we can neglect 

 the frictional resistances. 



First assume the distribution of pressure as given and imagine the 

 isobaric surfaces to become suddenly rigid and that velocities such as 

 are indicated by equation (11) are communicated to heavy points 

 mo vable without friction along these surf aces, then these will all remain 

 on the horizontal circles and move forward as before in similar 

 manner, since in this case there arises from the resistances of the sur- 

 face a force directed inward that holds in equilibrium the force gtgct 

 directed outward. 



In this case the acceleration in the direction of the slope that is 

 communicated by gravity to a heavy point resting on the surface is 

 g sin a, whereas the component of the outwardly directed horizontal 

 force gtg a that raises the point upward along this surface is g tan a 

 cos a, that is to say it is g sin a also. 



Whenever the velocities at any point whatever, or along any 

 horizontal circle whatever, become greater or smaller than those 

 required by equation (11) then the point will rise or fall respectively. 



Therefore these velocities given by this equation for any particular 

 distribution of pressure are called the "critical velocities," whereas 

 the isobaric surfaces given by this equation (n) for any particular 

 set of velocities will be called "critical surfaces." 



But the gradient corresponding to this critical distribution of 

 pressure will be known as the "critical gradient," in distinction from 

 the "effective gradient" ordinarily present, so that the fundamental 

 condition for the existence of a centered whirl may be expressed 

 thus: "In centered whirls the isobaric surfaces must coincide with 

 the critical surfaces and the effective gradients must equal the 

 critical gradients." 



By the help of this theorem we at once see that it is not at all prob- 

 able that a cyclone that is centered at the earth's surface should also 

 possess the same peculiarity at great altitudes. 



The distance between two isobaric surfaces is in general through- 

 out their whole extent subject to only moderate variations, since 

 it is simply proportional to the absolute temperatures prevailing at 

 the various locations. 



