120 ON THE MOTION OF SOLIDS THROUGH A LIQUID. [CHAP. V. 



dV dV 



which is readily found to be equal to f Trpa 3 -j- , or \ M f -*- , 



etc ctt 



if M denote the mass of fluid displaced by the sphere. 



If we suppose that the sphere started from rest under the 

 action of a force X constant in direction, so that the centre moves 

 in a straight line, the equation is 



or (M + \Wy~ = X (19). 



The sphere therefore behaves exactly as if its inertia were in 

 creased by half that of the fhlid displaced, and the surrounding 

 fluid were annihilated. 



We have assumed throughout the above calculation that the 

 motion of the sphere is rectilinear. It is not difficult to extend 

 the result to the case where the motion is of any kind whatever. 

 This is effected however more simply by the method of the next 

 article. 



The same method can be applied with even greater ease to the 

 case of a long circular cylinder, for which the value of $ was ob 

 tained in Art 86 . It appears that the effect of the fluid pressure 

 is in that case to increase the inertia of the cylinder by that of 

 the fluid displaced exactly. 



The practical value of these results, and of similar more general 

 ones to be obtained below, is discussed in note (E). 



106. The above direct method of calculating the forces exerted 

 by the fluid on the moving body would, however, in most cases 

 prove exceedingly tedious. This difficulty may be. avoided by a 

 method, first used by Thomson and Tait*, which consists in treat 

 ing the solid and the fluid as forming together one dynamical 

 system, into the equations of motion of which the mutual re 

 actions of the solid and the fluid of course do not enter. As a 



* Natural Philosophy, Art. 331. 



