514 SMITHSONIAN MISCELLANEOUS COLLECTIONS VOL. 51 



For a cyclone wherein the maximum absolute value of s is 3/76 

 we may estimate the average value [r] as being at the very highest, 

 only 3/76 2 as in the last example, since the smaller values of e cover 

 by far the larger surfaces. In order that K should equal A the 

 value of [G 2 ] must be(6.4) 2 , hence the average value of the velocity 

 must be not more than 6.4 meters per second. 



Observations give even for the lowest layer a greater average 

 velocity than this for the lowest barometric pressure of 73o mm . 

 If the radius of the cyclone is io° of a great circle and the average 

 gradient 3 mm mercury, then the mean wind velocity for the lower 

 stratum of air (about 20 meters above the ground) at median lati- 

 tudes is to be estimated at 12 meters per second, but that of all 

 the higher strata at, at least i8m/sec, and therefore K is at least 

 8 times larger than A. 



(5.) RELATIONS BETWEEN PRESSURE AND WORK 



The equation (I) has been based on the conception of work done 

 by the expansion of the gas and is provisionally spoken of as the 

 potential energy of the distribution of pressure within the closed 

 volume k. If this be correct then the work done by the pressural 

 forces in any elementary portion of time must be equal to the 

 change in —A. 



The work done by the pressural forces on the small mass dm dur- 

 ing dt is given by the expression 



- 5m . dp Gdt=-G dp dk.dt 

 /j. ds ds 



where ds is an element of the path of the moving mass and G is the 

 velocity in the direction ds. This is equivalent to 



iu dp + v dp + w dp )dk.dt 

 \ dx dy dz ) 



where u, v, w are the component velocities along the axes of x, y, z. 

 We have to prove that 



= f (u dp + v dp + W dp )dk. 

 J \ dx dy dz 



dA 

 dt J \ dx dy 



