.34 IRROTATIONAL MOTION. [CHAP. Ill 



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

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



The first part, whose components are u t 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 + by 2 + cz 2 + 2/yz + 2#zx + 2/txy = 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' = cY (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 f, 77, 



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

 liquid in which 



u = Zp.y, v=0, w=0, 



so that a, b, c, /, g, , ^ = 0, h = ^ =-/*. 



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. 



