238 Prof. Cliallis on the Principles of Hydrodynamics, 



be the axis of a;, we have for the components of the velocity at 

 any distance r from the centre, 



u=-^{x-u)% v=-^{ic-u)y, w=-^{w-c*)g. 



These values will be found to satisfy the equation (8,), r' being 

 equal to {x—aj^+y'^ + s'^. 



Again, the general equation of the surfaces of displacement is 



{x'-uf + y'^-^z^==a\ 



a and a being constant in passing from one point to another of 

 a given surface at a given time. Hence 



i|r = (a? — «)^ + y^ + ^^ — a^, 



But we have already proved (p. 34) that the function yfr has not 

 generally the same value for diffei*ent elements. Hence a and a 

 must be considered functions of t depending on the position of 

 a given element. Thus we have 



And because u=\-r-, it follows that 

 ax 



Since a particle in contact with the sphere remains in contact in 

 successive instants, we have for such a particle 



(l)=«'-'©=^- 



Also, as the velocity at the distance a, in any radius of the sphere 

 produced, is to the velocity at the point of intersection of the 

 sphere by the radius as R^ is to «^, it is clear that the motion of 

 any element at the surface of displacement whose radius is a is 

 the same as if a solid sphere of that radius moved foi-ward with 



R^ 



the velocity ^\'—^' Hence we have, generally, 



R2 



Consequently 



d'f 2ViR2 . , 



These values of the partial differential coefficients of ilr and the 



