24 G. I. TAYLOR ON EDDY MOTION IN THE ATMOSPHEEE. 



This expression is true for all disturbances, however large, but when the distance 

 z -z is small the first term only is of importance. Now it is evident that w is related 

 to z z by the relations 



"" 



Hence 



D 



or 



If w, w, and ZL, z are small, this is equivalent to 



Hence 



ff w(z a -z)dxdy = - U ff - (s-z) rZujffy- if (z <1 -z)^-(z l) -z)dxdy. 



J.JA JJA OX ^ J A C/C 



/ p ^ 



Now when a large area is considered I (z z) dx dy integrates out and vanishes. 



Hence 



w(z -z)dxdy= - jjj ^(z t) -z) 2 dxdy = - jJ| A (z.-z) 2 dxdy. 



It appears therefore that the rate at which the x-momentum in the slab A 

 increases is 



Integrating with respect to t we find that the difference between the momentum 

 in the slab A after and before the disturbance set in is 



Lord EAYLEIGH has pointed out that it is difficult to define instability. In the 

 present case the motion will be held to be unstable if the average value of the square 

 of the distance of any portion of the fluid from the layer out of which the disturbance 



