68 THE MECHANICS OF THE EARTH'S ATMOSPHERE. 



that where we have to do practically with the motions of fluids we are 

 thrown almost entirely back upon experimental trials, and can often, 

 from theory, predict but very little, and that only in an uncertain 

 manner, as to the result of new modifications of our hydraulic machines, 

 aqueducts, or propelling apparatus. 



In this state of affairs I desire to call attention to an application of 

 the hydro-dynamic equations that allows one to transfer the results of 

 observations made upon any fluid and with an apparatus of given 

 dimensions and velocity over to a geometrically similar mass of another 

 fluid and to apparatus of other magnitudes and to other velocities of 

 motion. 



To this end I designate by u v w the components of the velocity of 

 the 6rst fluid in the directions of the rectangular coordinate axes x y z; 

 by t the time, by p the pressure, by s the density, by k its coefficient of 

 friction (viscosity). The equations of motion in the Eulerian form in- 

 troducing the frictional forces, as is done by Stokes, in case no exterior 

 forces act upon the fluid, will now have the following form : 



<">*? 



dju.e) d(v.s) ftw.j) m 



jt dx T dy ~ r dz 



1 dp d« , ?U JU JU i tfu ?ll ?U * 



-TTx = Tt+ u ^+ v Ty+ w Tz- k \^+W + W> I 



_^i_ ) du dv dw i 

 3 dx \ 'dx + dy + lz ) ( la) 



To these are still to be added the two equations that are deduced from 

 the latter equation (la) by interchanging x and u with y and v or with 

 z and ic. 



When now for another fluid the velocities are designated by U, V, 

 W, the pressure by P, the coordinates by X, T, Z, the time by T, the 

 density by E, the viscosity constant by K, and if we introduce three 

 constants q, r, and n, and put 



K=qk (2) 



E=rs (2a) 



U~nu X=-x 



n 



V=nv . Y= ii 



n J 



W=niv Z=~z 



n 



P= n 2 rp + constant. T= %t 



then the quantities designated by these capital letters will also fulfill 

 the above differential equations. If we substitute these in those equa- 

 tions, the result, E, is as if all the terms of equation (1) were multiplied by 



