326 Prof. Challis on a Mathematical Theory 



But it is readily seen from the geometrical conditions, that 



_ ^ = W cos ^ and ''^ = W sin d. 

 at at 



Hence, substituting in the general equation, 



«VNap.logp+)^^;-+^=/(0, 



integrating from ?- = c to ?• = an indefinitely large value, and 

 supposing that /d = 1 at very great distances from the sphere, the 

 following equation will be obtained : — 



«^a^Nap. logp= — ^ccos^+ — cos20. ... (a) 



This equation gives the pressure at any point of the surface of 

 the sphere, when W is a given function of the time. 



It appears by this result that the pressure, so far as it depends 

 on terms of the second order, is the same at opposite points 

 of the sm-face of the sphere. Consequently the oscillations of 

 a small sphere in an elastic fluid are not aflfected by the pressures 

 depending on terms of the second order. The pressure indicated 

 by the term of the first order is the same with opposite signs at 

 opposite points of the sphere. Hence if W be expressed by a 

 periodic function, as m sin bt, the resultant of all the pressures 

 on the sphere will be a periodic pressure which vanishes when 

 W is a maximum, and is greatest when W = 0. 



If the sphere move uniformly in the fluid, —7— = 0, and the 



resultant pressure upon it is zero. Hence a very small sphere 

 moving uniformly in a very elastic fluid, suffers no retardation, 

 so far, at least, as terms to the second order indicate. This 

 result I have employed (Phil. Mag. for May 1859), on the hy- 

 pothesis that the constituent atoms of bodies are minute spheres, 

 to account for the fact that the resistance of the setherial medium 

 to the motions of planets is of inappreciable magnitude. 



Problem IV. A sei'ies of waves defined by the equation 

 'W = Kaa are incident on a small sphere at rest: it is required 

 to find the pressure at any point of its surface. 



The pressure on the half of the surface on which the Waves are 

 directly incident requires to be considered separately from the 

 pressure iu directly produced on the other half. 



It will be supposed that the diameter of the sphere is extremely 

 small compared to \ the breadth of an undulation, and that 

 the variations of velocity and condensation at a given instant, 

 through a space equal to the sphere's diameter, are so small that 

 they may be neglected. Hence, W and cr always representing 



