38.] ANALYSIS OF MOTION OF AN ELEMENT OF FLUID. 83 



Hence the motion of a small element having the point (an, y, z) for 

 its centre may be conceived as made up of three parts. 



The first part, whose components are u, v, w, is a motion of 

 translation of the element as a whole. 



The second part, expressed by the second, third, and fourth 

 terms on the right-hand side of the equations (2), is a motion such 

 that every point on the quadric 



aX* + b Y* + cZ* + 2/FZ+ 2gZX + 2hXY= const (3), 



is moving in the direction of the normal to the surface. If we 

 refer this quadric to its principal axes, the corresponding parts of 

 the velocities parallel to these axes will be 



u f =a X t v = vr, w = cz f (4), 



where a X 2 + b Y 2 -f c Z 2 = const. 



is what (3) becomes by the transformation. The formulae (4) express 

 that the length of every line in the element parallel to X is being 

 elongated at the rate (positive or negative) a , whilst lines parallel 

 to Y and Z are being similarly elongated at the rates 6&quot; and c 

 respectively. Such a motion is called one of pure strain or dis 

 tortion. The principal axes of (3) are called the axes of the strain 

 or distortion. 



The last two terms on the right-hand side of the equations (2) 

 express a rotation of the element as a whole about an instan 

 taneous axis ; the component angular velocities of the rotation 

 being f , 77, f . 



It can be shewn that the above resolution of the motion is 

 unique. If we assume that the motion relative to the point (x, 

 y, z) can be made up of a distortion and a rotation in which the 

 axes and coefficients of the distortion and the axis and angular 

 velocity of the rotation are arbitrary, then calculating the relative 

 velocities Uu t V- v, W w, we get expressions similar to those 

 on the right-hand side of (2), but with arbitrary values of a, 6, c, 

 f&amp;gt; 9&amp;gt; h, t;, 77, ?. Equating coefficients of X } Y, Z, however, we find 

 that a, b, c, &c. must have the same values as before. Hence the 

 directions of the axes of distortion, the rates of extension or con 

 traction along them, and the axis and the angular velocity of rota- 

 L. 3 



