120 Mb. stokes, ON SOME CASES OF FLUID MOTION. 



the efFect of the inertia of the fluid will be to inci-ease the mass of the sphere by a mass 



f^ la + — \ 



3 6' -an 2aV 6' - o= 2 



H being the mass of the fluid displaced; so that the effect of the case is, to increase the mass 

 which we must suppose added to that of the ball in the ratio of 6' + 2a^ to 6^ — al 



Poisson, in his solution of the problem of the oscillating sphere given in the Memoires de 

 rinstiiiit. Tome xi. arrives at a different conclusion, viz. that the case does not at all affect 

 the motion of the sphere. When the elimination which he proposes at p. 563 is made, the last 



term of equation (f) p. 550 becomes — -^ 5 — - (-r4H ^ , where a is the velocity of 



^ •' '^ 2a'c\{l - Sy) \dt' df ) ^ 



propagation of sound, and S the ratio of the density of air to that of the ball, ^ and t' being 



functions derived from others which enter into the value of d) by putting r = c, where c is the 



radius of the ball. He then argues that this term may be neglected as insensible, since it involves 



S in the numerator and a" in the denominator, tacitly assuming that — f + — ^ is not large 



since <p is not large. Now for the disturbances of the air which have the same period as 



those of the pendulum ~ is not large compared with cf>, as it is for those on which sound 



depends. Let then Poisson's solution of equation (a), p. 547 of the volume already mentioned, 

 be put under the form 



/=M/('-3-(-3hi{/('-3--(-3). 



/' and F' denoting the derived functions, and all the Laplace's coeflficients except those of the 

 first order being omitted, the value of <b just given being supposed to be a Laplace's coefficient 

 of that order. Then if we expand the above functions in series ascending according to powers 



of — , we find 

 a 



9 = ^. !/(') + ^(0} - ^ {/" W + F"m + ^ {/'"(O - F-(0\ + ... ; 



and in order that when a = es this equation may coincide with (10), when all the Laplace's 

 coefficients except those of the first order are omitted in that equation, it will be seen that it is 

 necessary to suppose f"'(t)-F"'(,f), and therefore f(f) - F(t), to be of the order «^ while 

 f(t) + F(i) is not large. Putting then 



f(t) = x(0 + «V(0, 



/'(0 = x(0-«V(0, 



we shall have 



dm'+r) 



so that — yi:r '*''ll contain a term of the order a", and the term which Poisson proposes to 



leave out will be of the same order of magnitude as those retained. 



In making the experiment of determining the resistance of the air to an oscillating sphere, it 

 would appear to be desirable to enclose the sphere in a concentric spherical case, which would at the 

 same time exclude currents of air, and facilitate in some measure the experiment by increasing the 

 small quantity which is the subject of observation. The radius of the case however ought not to be 

 nearly as small as that of the ball, for if it were, in the first place a small error in the position of the 



