360 Prof. Challis on the Motion of a small Sphere 



requires the application of new principles, I do not profess to 

 have solved. I have elsewhere stated that motion compounded 

 of direct and transverse vibrations relative to an unlimited num- 

 ber of parallel axes might be such as to be contained within a 

 cylindrical surface of small radius; but I have not attempted to 

 determine the law of the diminution of the motion at parts con- 

 tiguous to the boundary. For this reason I am not prepared to 

 determine the effect of the transverse action in the present pro- 

 blem so as to obtain expressions for the velocity and condensa- 

 tion at all positions beyond the plane through the centre of the 

 sphere. Still it is possible to arrive at certain results of much 

 theoretical importance, which I now proceed to develope. 



Considering only positions contiguous to the second hemisphe- 

 rical surface, the state of the fluid there is required to fulfil the 

 following three conditions : (1) it must be such as to satisfy the 

 equation (77) ; (2) the motion being wholly along the surface, 

 U = 0; (3) as the transverse action affects the amount but not 

 the phase of the condensation, the change of condensation it pro- 

 duces at a given point varies as <r v or rrJ sin q(a't-hc ). Now 

 these conditions may all be fulfilled by means of the foregoing 

 integration involving the function <p q , if only the arbitrary con- 

 stants be differently determined. For by that integration we 

 may have 



dK "*" iqk3 ) ( m 5 s * n y a '* ~*~ m, 5 cos Q a, t) cos2 Q- 



Hence -7- =0, and consequently U=0, where r = c. Let 



m 5 =hm 3 and m'^==km' 3 , m 3 and m' 3 being the former values of 

 the constants, and h being an unknown constant depending on 

 the transverse action. Then putting c for r, substituting the 

 values of m 3 and m! 3 , and subtracting the resulting equation from 

 the former value of <T-—cr l} we shall obtain, for the change of con- 

 densation produced at the second hemispherical surface by the 

 transverse action, 



5o 2 c 2 

 jr-j- (1 —h)m! sin q(a't -f c ) cos 2 0, 



which satisfies the third condition. This expression vanishes if 



77* 

 0= — as also does the corresponding expression for the velocity. 



The whole pressure on the sphere due to this condensation and 

 estimated in the direction of propagation, will be found, by inte- 



rrr • 



grating from 6= jr to = 7r, to be 



12a! (l-*Xs»i0(ffl'f + c o ). 



