36 



TBE MECHANICS OF THE EARTHS ATMOSPHERE. 



in (Ic) compounded only of a motion of translation in space and an ex- 

 pansion and contraction of its edyes and it has no rotation. 



We return now to the first system of coordinates, that of a-,?/, z, and 

 imagine added to the hitherto existing motion of the infinitely small 

 mass of liquid surrounding the point i;, i), ], a system of rotatory motions 

 about axes that are parallel to those of a", y, s, and that pass through 

 the point r,, i), 3, and whose angular velocities of rotation may be ^, ;/, C, 

 thus then the component velocities parallel to the coordinate axes of 

 x, y, z, as resulting from such rotations are respectively : 



Therefore the velocities of the particles whose coordinates are x, ?/, z, 

 become: 



u=A + a{x - r,) + ( y + Z) [ij - l)) + (/J - 7/) (z-s), 



r=B+ {y-Z} ix-v.) + h{v-y) + {a + S) {z-',), 



U'=C+ (/?+?/) {x-v) + (a— <?) (j/-i)) + c(c-^), 



whence by differentiation there results : 



iJz ,yy 



(2) 



i-^ __ O^ __ f)P- 



3 



Therefore the quantities on the left-hand side, which according to 

 equation (Ic) must be equal to zero in order that a velocity potential 

 may exist, are equal to double the velocity of rotation about the three 

 coordinate axes of the liquid particles under consideration. The exist- 

 ence of a velocity potential excludes the existence of a. rotary motion 

 of the particles of liquid. 



As a further characteristic peculiarity of fluid motions that have a 

 velocity potential, it may be further stated that in a simply-connected 

 space Sj entirely inclosed within rigid walls and wholly filled with 

 fluid, no such motion can occur ; for when we indicate by n the nor- 

 mal directed inwards to the surface of such space then the component 



velocity ~ directed perpendicular to the wall must be everywhere 



