36 



THE MECHANICS OF THE EARTH'S ATMOSPHERE. 



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

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



We return now to the first system of coordinates, that of %,y, 0, and 

 imagine added to the hitherto existing motion of the infinitely small 

 mass of liquid surrounding the point 5, 1), j, a system of rotatory motions 

 about axes that are parallel to those of a?, y, 0, aud that pass through 

 the point 1;, 1), 5, and whose angular velocities of rotation may be £, 77, C, 

 thus then the component velocities parallel to the coordinate axes of 

 x, y, 0, as resulting from such rotations are respectively : 



Parallel to x\ 



o. 



-(s-5) 7> 



Parallel to y : 



o, 



-(#-£) C, 



Parallel to 0: 

 o. 



Therefore the velocities of the particles whose coordinates are x, y, z, 

 become : 



u=A + ct(x - E )+( r + (2/ - i)) + (/5 - 7) (*-i)» 



m,= C+ {0+t?) (a?- J) + («-£) (y-t))+c(0-j), 

 whence by differentiation there results : 



J <5« 



3«J ^M? 



dz'W 



?y ?x fc, _J 



(2) 



Therefore the quantities on the left baud side, which according to 

 equation (lc) 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 uuder 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 8, 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 l directed perpendicular to the wall must be everywhere 



