EXERCISES. 



1. If the motion of a fluid in two dimensions be referred to polar 

 co-ordinates r, 6 t and if u, v denote the component velocities along and 

 perpendicular to the radius vector, the component accelerations in the 

 same directions are 



, dv dv dv uv 



and + u -=- + v -- + , 



dt dr rdO r 



respectively. 



2. If in the solution of the equations 



dx dy dz 



Tt= w lu = v dt = w 



where u, v, w are given functions of x, y, z, t, satisfying the condition 



du dv dw 



x, 2/5 be expressed as functions of t, and the arbitrary constants a, b, c, 



d (x v z\ 



the determinant . , _ - is independent of t. 

 d (a, 6, c) 



3. If F (x, y, z, t) = be the equation of a moving surface, the 

 velocity of the surface normal to itself at any point is 



1 dF fdF\ 2 fdF\* fdF\ 2 



- T&amp;gt; -77 , where A~= ) +(^- + { . 



R dt \dx) \df/J \dzj 



Hence deduce the surface-condition of Art. 10. [Thomson.] 



