Attractive and Repulsive Forces, 197 



centre of the sphere, and let 5 be any length along this line 

 reclvoned from a given point on it as origin. The sphere being 

 by hypothesis extremely small, it may be assumed, in accordance 

 with what is argued above, that for all points of any transverse 

 circular area the centre of which is on the trajectory, and the radius 

 not less than the radiusof the sphere, witli sufficient approximation 

 p=f{s). Let 5y be the value of 5 corresponding to the spherc^s 

 centre, and let 6 be the angle which any radius of the sphere 

 makes with the trajectory, so reckoned that for a point of the 

 surface 5 = 5y — 5 cos ^, b being the radius. Then for any such 

 point 



P=A^i~^ cos 0) =f{s;)—b cos 6f'{sj) very nearly, 

 the remaining terms being omitted because the variation of p 

 through the small extent of the sphere's diameter maybe assumed, 

 with sufficient approximation, to be uniform. Accordingly the 

 whole pressure on the sphere estimated in the positive direction 

 of 5 is 



27T ^a^pb'' sin eco& 6 d6, from 6 = to (9 = 7r. 



This integral, on substituting the above value of jO, will be found 

 to be 



Hence, A being the density of the sphere, the accelerative force 

 is —-T-f'i^i)' ^^ Pi ^^^^ ^i ^® ^^c density and velocity corre- 

 sponding to the centre of the sphere, 



,,=/(,)=p„(l-g)and/'(,) = -^J.^^. 



Hence, by substituting for/'(5y), the expression for the accelera- 

 tive force becomes 



Po.^idY, 



A dsi 



This result proves that the accelerative action on the fixed sphere 

 has a constant ratio to the acceleration of the fluid where the 

 sphere is situated, and is in the same direction. The direction 

 is therefore positive or negative according as p decreases or in- 

 creases, or according as V^ increases or decreases as s increases. 



20. If the sphere, instead of being fixed, moved uniformly in 

 a given direction, the accelerative action of the fluid upon it in 

 any position would still have the same constant ratio to the ac- 

 celeration of the fluid in the same position. For if the uniform 

 motion be impressed in the contrary direction both on the fluid 

 and the sphere, the latter will be reduced to rest, and the rehw 



