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Full text of "Scientific Papers - Vi"

Here x, y are the coordinates of a point, referred to axes revolving uniformly in the plane xy with angular velocity &>*, u .and v are the components of relative velocity of the fluid in the directions of the revolving axes, that is the components of wind. We have now to define the motion for which we wish to determine the balancing pressures.
We contemplate a motion (relatively to the ground) of rotation about a centre C, Fig. 1, situated on the axis of x, the successive rings P at distance
Pig. 1.
R from C revolving with an angular velocity ", which may be a function of R. And upon this is to be superposed a uniform velocity of translation U, parallel to x and carrying everything forward. If initially 0 be at 0, the fixed origin, its distance from 0 along Ooc at time t will be Ut. Thus
u = If  <y(   v = %(x  Ut\ ........................(4)
 being a known function of R, where
These equations give u and v in terms of the coordinates and of the time, and the values are to be introduced into (1) and (2). From the manner in which as and t enter (representing a uniform translation of the entire system) it is evident that djdt =  Ud/doc. We have
du___%'Xy      du
dx          R
____r _2_M
dy~    ^     R~
__  f 4. dx~^^
being written for dtydR ; and
Dt        R
_ Dt
* In the application to a part of the earth's atmosphere, w is the earth's angular velocity multiplied by the sine of the latitude. starting from the usual Eulerian equations as referred to uniformly rotating axesf. The density (/D) is supposed to be constant, and gravity can be disregarded. In the usual notation we have