48 THE MECHANICS OP THE EARTH'S ATMOSPHERE. 



to these particles such a mean velocity of progression parallel to the 

 surface as corresponds to the arithmetical mean of the velocities 

 [v l and v x \ prevailing on the two sides of the layer. 



For instance such a vortex sheet would be formed when two fluid 

 masses previously separated and in motion come into contact with 

 each other. At the surface of contact the velocities perpendicular 

 thereto must necessarily balance each other. l\\ general the velocities 

 tangent to this surface will, however, be different from each other in 

 the two fluids. Therefore the surface of contact will have the prop- 

 erties of a vortex sheet. 



On the other hand, we should not in general think of individual 

 vortex filaments as infinitely slender, because otherwise the velocities 

 on opposite sides of the filament would have infinite values and oppo- 

 site signs, and therefore the velocity proper of the filament would be 

 indeterminate. In order now to draw certain general conclusions as to 

 the movement of very slender filaments of any sectional area, the prin- 

 ciple of the conservation of living force will be made use of. 



Therefore before we pass to individual examples, we must first write 

 the equation for the living force K of the moving mass of water, or 



K=$h\ \(u 2 +v 2 +w 2 )dxdyds. (6) 



In this integral I substitute from equation (4) 



u 2 =u( -T- + -, — ) 



\dy^dz dxj 



W Z =W[ 7T-+- -~ ) 



v, dz dx dy J 



and integrate by parts ; then I indicate by cos a, cos /5, cos y, and cos 6 

 the angles made by the coordinate axes and the resulting velocity, q, 

 respectively with the interior normal to the element doo of the mass 

 of liquid and having regard to equations (2) and (1 4 ) I obtain: 



A= - 2 'M-Ptf cos #+L(v cosy— wcos/?) 

 + M(w cos a— u cos y) + N(u cos fi— v cos a)\ (Qa) 



{hZ+Mrf+lSQdx dy dz. 



The value of 



d_K 

 dt 



