ATOM. 467 



surface. Lastly, it is a perfect fluid, or, in other words, the stress between 

 one portion and a contiguous portion is always normal to the surface which 

 separates these portions, and this whether the fluid is at rest or in motion. 



We have seen that in a molecular fluid the interdiffusion of the molecules 

 causes an interdiffusion of motion of different parts of the fluid, so that 

 the action between contiguous parts is no longer normal but in a direction 

 tending to diminish their relative motion. Hence the perfect fluid cannot be 

 molecular. 



All that is necessary in order to form a correct mathematical theory of 

 a material system is that its properties shall be clearly defined and shall be 

 consistent with each other. This is essential ; but whether a substance having 

 such properties actually exists is a question which comes to be considered 

 only when we propose to make some practical application of the results of the 

 mathematical theory. The properties of our perfect liquid are clearly defined 

 and consistent with each other, and from the mathematical theory we can 

 deduce remarkable results, some of which may be illustrated in a rough way 

 by means of fluids which are by no means perfect in the sense of not being 

 viscous, such, for instance, as air and water. 



The motion of a fluid is said to be irrotational when it is such that if a 

 spherical portion of the fluid were suddenly solidified, the solid sphere so formed 

 would not be rotating about any axis. When the motion of the fluid is rota- 

 tional the axis and angular velocity of the rotation of any small part of the 

 fluid are those of a small spherical portion suddenly solidified. 



The mathematical expression of these definitions is as follows : Let u, v, w 

 be the components of the velocity of the fluid at the point (x, y, z), and let 



dv dw ,._dw du _du _dv^ , . 



a = dz~Jy' P = ~dx~dz' 7 ~d^~dx- 



then a, , y are the components of the velocity of rotation of the fluid at the 

 point (x, y, z). The axis of rotation is in the direction of the resultant of 

 a, ft, and y, and the velocity of rotation, (a, is measured by this resultant. 

 A line drawn in the fluid, so that at every point of the line 



__^ = 

 a ds ~ ft ds ~ y ds <a 



where s is the length of the line up to the point x, y, z, is called a vortex 



592 



