352 PROFESSOR CHALLIS, ON THE MOTION OF A SMALL SPHERE 



or, putting (p for <p (s, y, as, t), 



<t> + -*£ . udt + -S- . vdt + ~- . wdt + -T7 dt = 0. 

 r dx dy dz dt 



Now = 0, -j^ = Nu, -j- = Nv, -S = Nw. Hence 

 r dx dy dz 



N(u* + v° + w 2 ) + S = 0. 



Or, if u* + r" + lb" =V\ N = - ^- f . 



Let, for example, the surface of displacement be that of a sphere of 

 given radius moving with a given velocity V t in the direction of the axis 

 of %. Then if R be the radius of the sphere, and a, b, c, be the co- 

 ordinates of its centre, 



(*, y, %, f) = {x - a)' + (y- bf + (u - cf - R\ 



rr dc d(b ', . dc „ rr , . 



v = d-r di = -^ % - c ^r-^ v ^-^ 



and the normal velocity Vis equal to V r _ . Consequently N = j^-. . 



Si r t \% — C) 



It therefore appears that the surface of an oscillating sphere may be a surface 



of displacement, and that the factor JV varies as g » as I have supposed 



in Art. 5. It also appears that the error of Poisson's solution consists in his 

 employing an equation depending on the supposition that udx + vdy + wdz 

 is of itself an exact differential ; a supposition which, as we have seen, is 

 not of sufficient generality. Indeed it would not be difficult to shew that 

 this condition is fulfilled only when the surfaces of displacement coincide 

 with surfaces of equal pressure during the whole of the motion, and when 

 in consequence the motion of each particle of the fluid is rectilinear. The 

 differential equations of fluid motion in their most general form have never 

 yet been obtained. 



The above considerations lead to a very simple solution of the problem 

 of the resistance of the air to an oscillating sphere. For supposing the 

 motion of the sphere to be impressed on the sphere and on the air in the 



