103 105.] PRESSURE OX MOVING SPHERE. 119 



surface of the solid. The most obvious way of doing this is, first 

 to calculate p from the formula 



* = const. _**-J(vel.) ................. (18), 



and then to find the resultant force and couple due to the pressure 

 p acting on the various elements dS of the surface by the ordinary 

 rules of Statics. We will work out the result for the simple case 

 of the sphere, starting from the value of &amp;lt;f&amp;gt; given by (12). Since 

 the origin to which &amp;lt;*is there referred is in motion parallel to OX 

 with velocity F, whereas in (18) the origin is supposed fixed, we 



must write, instead of - , 



_ 



dt &quot; dx 



where x r cos 6. Now 



d6 dd&amp;gt; * d . TV 



and 



The whole effect of the fluid pressure evidently reduces to a force 

 in the direction OX. The value of p at the surface of the 

 sphere is 



dV 



p = const. + ^pa -=- cos + &c., 

 dt 



the remaining terms being the same for surface-elements in the 

 positions 6 and TT 6, and therefore not affecting the final result. 

 Hence if V be constant, the pressures on the various elements 

 of the anterior half of the sphere are balanced by equal pressures 

 on the corresponding elements of the posterior half ; but when the 

 motion of the sphere is being accelerated there is an excess of 

 pressure on the anterior, and a defect of pressure on the posterior 

 half. The reverse holds when the motion is being retarded. The 

 total effect in the direction of V, is 



r* 



%7ra sin . adO . p cos 0, 

 Jo 



