4-64. Rev. J. Challis on the Motion of a small Sphere 

 and passing from differences to differentials, 



Tt + t*dr = 5 ..... (h) 



where, from the nature of the investigation, the differential 

 coefficients are evidently partial. 



Now suppose the motion of the fluid to be directed to or 

 from a moving centre, and let two spherical surfaces separated 

 by a very small interval be described about this centre, the 

 interior one always passing through the point of space at 

 which we consider the motion. On account, therefore, of the 

 motion of the centre, the spherical surfaces will not be sta- 

 tionary. We may i( however, conceive a conical surface, de- 

 scribed as in the former case, to have its axis always passing 

 through the moving centre and the point of space at which 

 the motion is considered, and to include a given small por- 

 tion m* of the interior spherical surface. The velocity and 

 density of the fluid passing the area m* may, as before, be 

 considered uniform during a very small time St; as may also, 

 without entailing error, the velocity and density of the fluid 

 passing the portion of the outer surface always included by 

 the conical surface. Hence, using the same letters as in the 

 case of a fixed centre, the quantity of fluid which passes m a 

 in the time B t is m 2 p v ot. We have now to ascertain the quan- 

 tity of fluid which in the same time passes the corresponding 

 area of the exterior surface. Let r and r' be the radii of the two 

 concentric surfaces at the beginning of the interval /, and 

 let be the velocity of the centre resolved in the direction of 

 r. Then after an interval T, less than St, the radii of thre 

 surfaces are r + T and t j + T ultimately. Hence the area 



of the outer surface corresponding to m 2 = m 2 . ( ~ " T ] 



\ r a. T ) 



/I + -\ 

 f - r \ 

 . 1 - I = 



m* r 12 , 



g ' "y neglecting terms that 



r 



may be neglected, since by hypothesis r 1 differs very little 

 from r, and a T is very small. This result is independent of 

 T, and is the same as if the centre had been fixed. The rest 

 of the reasoning would consequently conduct to the equation 

 (1.). Hence from this equation combined with the known 



equations p (the pressure) = a 2 p, and j+ (~\= 0, 

 equations applicable to motion directed to or from either a 



