PEIRCE. — OSCILLATIONS OF SWINGING BODIES. 65 



velocity, would encounter no resistance from the liquid if there were 

 no viscosity ; but that even in a homogeneous, perfect liquid, a sphere 

 moving with changing velocity would meet with a resistance from the 

 liquid, and the inertia of the sphere would in consequence of this be 

 apparently increased in a manner which could be mathematically 

 accounted for in the equation of motion of the sphere, if the mass of 

 the sphere were increased by half the mass of the displaced liquid. 



If at the point (x, y, z) in a viscous fluid at the time t the components 

 of the velocity are u, v, w, if the applied body forces which urge the 

 fluid have the components X, Y, Z, if p is the density, and if fi repre- 

 sents a constant of the fluid which measures its coefficient of viscosity, 

 the equations of motion of the fluid as established by Navier and 

 Poisson 5 are usually written in the forms : 



(du , du du du\ dp dm 



(dv dv dv dv\ v dp dm 



P \dJ + U -rx + V -Ty + W -d- Z ) =pY -dy + ^-dy +IX -^> 



(dw dw dw dw\ dp dm 



p {w +U -dx +V -dy +W -dj) = pZ -^ + ^--dz- + fX ' V(w) ' 



(1) 

 . du , dv dw 



where m = ^ — h w~ + -~-, 



ex oy Cz 



and p represents the arithmetical mean of the normal pressures on any 

 three mutually perpendicular planes through the point (x, y, z). 



Using these equations, Stokes, in a paper 6 presented to the Cam- 

 bridge Philosophical Society in December, 1850, determined the 

 resistance which a sphere making small harmonic oscillations of com- 

 plete period T, in an infinite viscous liquid, would encounter, and 

 showed that if 6 represented the distance of the centre of the sphere 

 from its mean position at the time t, the value of this resistance would 

 be 



{2 + 4aj) M '-d¥ + 2ajTV + af)-di> (2) 



,,,. d 2 9 d9 . 



M" w + 2m s , (3) 



5 Navier, Memoire de l'Academie des Sciences, 6, 1822. Poisson, Journal 

 de l'Ecole Polytechnique, 13, 1829. 



6 Stokes, Mathematical and Physical Papers, II. 

 vol. xliv. — 5 



