IN HYDRODYNAMICS. 41 



df df df 



/ = 2 Cr - a), / = 2t/, -f = 2;jr. 



da- ' dy dz 



F(a;-a) 



We have now to find F by means of equation (8). Since the above value of A'' shews that 

 udx + vdy + wdz is not for this instance an exact differential, V must be differentiated along 



a line of motion. Hence putting cos for , we have V= F cos 0, and d F = d F cos0; so 



a 



, dV dV ., , , 112 ^ . / V . ^ , dF 2da 



that — = . Also ds = da and - + - = -. Equation (8) therefore becomes — — + = 0. 



V V^ r r a ^ ' F a 



f(t) f(t) 



Hence F = -—^ , and F="^-^cos0. Thus the motion is completely determined. It is plain 

 ' a- a- 



that this motion would be produced by a smooth solid sphere moving in an arbitrary manner in the 



fluid, with its centre always in a given straight line. 



Ex. 4. Poisson's determination of the simultaneous motions of a sphere and the surrounding 

 fluid (Memoirs of the Paris Academy, Tom. xi, and Connaissance des Tems, An. 1834) differs 

 from the foregoing. Let us therefore inquire, assuming the motion to be such as Poisson has 

 found, whether the conditions of continuity are satisfied. 



For the sake of simplicity I shall consider the fluid to be incompressible. Poisson assumes 

 that udx + vdy + wdz = d(p, and finds values of the velocities which, if i?^ = (x — aY + y'' + z^, 

 may be thus expressed : 



Tc' f 3(x-ay\ 3Tc^ , , STc" , 



T being an arbitrary function of the time, and c the radius of the sphere. These values make 



udx + vdy + wdz an exact differential, and satisfy the equation — ^h —+ — v= 0. By 



da^ dy^ dz' •' 



integrating udx + vdy + icdx =0, the equation of the surfaces of displacement will be found 

 to be R^- H' (,» - a) = for the positive values of x - a, and R^ + h^ {x - a) = for the negative 

 values. This change of equation implies a breach of continuity*. If we put R cos 6 for 

 X - a, we obtain R^—h?cos6 = for the polar equation of the curve which by its revolution 

 about the axis of x generates a surface of displacement. The lines of motion lie in planes 

 passing through the axis of x. The general polar equation of these lines will be found to be 

 R-esm''6 = 0. From the latter of these equations the value of ds is to be found, and from 

 the other the values of r and r', for the purpose of ascertaining whether the equation (8), 



. dF /I i\ . 



VIZ. — + d« |- + -1 = 0, is satisfied by these values so as to allow of its being integrated 



between arbitrary limits, the dynamical equation having been already integrated in this manner. 



• According to this »olution the fluid in contact with the 

 •phere and in a plane passing through its cenlrc perpendicular 

 to the axis of j, movtn backwards willi half the velocity with 



Vol. VIII. Paut I. 



which the sphere moves forwards, a result, to say the least, 

 very Improbable. 



