34 IRROTATIONAL MOTION. [CHAP. Ill 



Hence the motion of a small element having the point (x, 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 first three terms on the 

 right-hand sides of the equations (2), is a motion such that every 

 point is moving in the direction of the normal to that quadric 

 of the system 



ax 2 + 6y 2 + cz 2 + 2/yz + 2#zx + 2/ixy = const (3), 



on which it lies. If we refer these quadrics to their principal axes, 

 the corresponding parts of the velocities parallel to these axes will be 



u = aV, v = 6 y , w = c z (4), 



if a x 2 + 6 y 2 + 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 (positive or negative) rate a, whilst lines parallel 

 to y and z are being similarly elongated at the rates b and c 

 respectively. Such a motion is called one of pure strain and the 

 principal axes of the quadrics (3) are called the axes of the strain. 



The last two terms on the right-hand sides 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 rj, 



This analysis may be illustrated by the so-called laminar motion of a 

 liquid in which 



u = 2py, v=0, w = 0, 



so that a, b, c, /, g, |, ,7 = 0, h = p, =-/*. 



If A represent a rectangular fluid element bounded by planes parallel to 

 the co-ordinate planes, then B represents the change produced in this in a 

 short time by the strain, and C that due to the strain plus the rotation. 



