134-135] MOTION OF A SPHERE. 205 



135. As a first application of the foregoing theory we may 

 take an example given by Thomson and Tait, where a sphere is 

 supposed to move in a liquid which is limited only by an infinite 

 plane wall. 



Taking, for brevity, the case where the centre moves in a 

 plane perpendicular to that of the wall, let us specify its position 

 at time t by rectangular coordinates a;, y in this plane, of which y 

 denotes the distance from the wall. We have 



2T=Ax* + By* ........................ (1), 



where A and B are functions of y only, it being plain that the 

 term xy cannot occur, since the energy must remain unaltered 

 when the sign of x is reversed. The values of A, B can be 

 written down from the results of Arts. 97, 98, viz. if in denote the 

 mass of the sphere, and a its radius, we have 



(2), 

 J 



approximately, if y be great in comparison with a. 

 The equations of motion give 



(3} 



J / J A JT> ' ^ '* 



a x-o.x -, fdA . . aB ., 



where X, T are the components of extraneous force, supposed to 

 act on the sphere in a line through the centre. 



If there be no extraneous force, and if the sphere be projected 

 in a direction normal to the wall, we have x = 0, and 



J$if- = const (4). 



Since B diminishes as y increases, the sphere experiences an 

 acceleration from the wall. 



Again, if the sphere be constrained to move in a line parallel to 

 the wall, we have y = 0, and the necessary constraining force is 



*if 



Since dA/dy is negative, the sphere appears to be attracted by the 



