SHORT MEMOIRS ON METEOROLOGICAL SUBJECTS. 449 



ated that, by moving from the level surface a a^ hi /3 upward parallel with 

 the axis of rotation CZ, we intersect the entire series of level surfaces 

 which is formed in the rotating volume. For each of these level sur- 

 faces, the pressure upon a unit of surface is constant ; but when we go 

 from one level surface to the next, the pressure diminishes as we go 

 upward, while it increases as we go downward toward the ground 

 plane XY. 



In order to prove the correctness of this statement, we will consider 

 an element of the rotating fluid, which is at the distance r from the 

 axis of rotation, and which moves with a velocity of v. If we consider 

 the direction C Z as positive, we shall, as is well known, have, since ^ 

 indicates the pressure upon a unit of surface for the element in ques- 

 tion, 



dp = p(' — (j dz + J^dr^ , 



where p indicates the density of the fluid and g the force of gravity. For 

 each of the level surfaces of the fluid, the pressure p is constant, and 

 dj) = 0, therefore 



fj dz =1 dr, or ^ = J_ (2) 



•^ r ' dr (j.r ^ ^ 



independently of the density p. From this equation, together with 

 equation (1), are deduced, directly, the above-stated propositions con- 

 cerning the level surfaces of the rotating volume of water. If the 

 expression for the current's velocity v, as given by equation (1), is sub- 

 stituted in equation (L*), and this equation is then integrated by making 

 r = a-\- X, and, for the sake of brevity, 



X=[l-0.188(-^y]„atlog(l + ?) 



-1.732(^)^,-.7i+0..77j(3j-|) 



+ 



we obtain 



z = Zo + ^\ J, (3) 



in which the arbitrary constant Zq represents the value of z foTx = 0, 

 so that formula (3) is the equation for any plane surface in the rotating 

 fluid.* 



If we now consider formula (1), which is applicable to water, together 

 with formula (3), which is based upon the same, while formula (2) is of 

 general application, we shall see that in a rotating volume of water the 

 velocity of the rotation increases from the exterior surface of such vol- 



* Note by the Translator.— The above formula) neglect tbe influence of the rota- 

 tion of the earth, and are therefore applicable strictly only to tornadoes and small 

 cyclones, or to cyclones within the tropics. — C. A. 

 20 S 



