Medium to the Motions of Planets and Comets. 355 



Cambridge Philosophical Transactions, vol. vii. part 3, in a paper 

 " On the Motion of a small Sphere in an Elastic Medium," 

 art. 7. In the former of these solutions the conclusion arrived 

 at is thus stated : — " When a small sphere moves in a medium 

 like air with a velocity small compared to the velocity of propa- 

 gation in the medium, and in any manner except in rapid vibra- 

 tions, the pressure on its surface is at every point very little 

 different from the pressure of the medium at rest." Having 

 made these references, I think it unnecessary to adduce the ana- 

 lytical reasoning in detail : it may suffice for the present purpose 

 to indicate the principles on which it proceeds, which are as fol- 

 lows. In an article "On the Central Motion of an Elastic 

 Fluid," contained in the Philosophical Magazine for January 

 1859, I have obtained the following general values of the velo- 

 city V and condensation S at the distance r from the centre at 

 any time t : — 



V=— 2.-( mcos—-{r + Kat + c) >, 



1 „ fmX . 27r , _ .\ 

 ^S. -^ g— sm — (r + /cfl^ + c) j-, 



>.2' 



«aS 



= — S . s + m cos — (r + Kut + e) r ■ 



The ratio of the coefficients of corresponding terms in the 



27n" 



two groups of terms that make up the value of V is r — , 



A. 



which, for a given value of r, is smaller as \ is greater and the 



vibrations less rapid. Hence as V approaches to uniformity for 



a given value of r, this ratio becomes indefinitely small, and the 



first group of terms vanishes in comparison with the other. 



Hence also the value of «S is of insensible magnitude ; and the 



effective pressure, which varies as a^S, may be exceedingly small 



notwithstanding the large factor a, when, in consequence of the 



motion being nearly uniform, the ratio ^ must be exceedingly 



small for a given small value of ?•. If the fluid be supposed to 

 be disturbed by the motion of a small sphere, the velocity im- 

 pressed by the sphere is at each instant directed to or from its 

 centre, because, being perfectly spherical, it is incapable of impres- 

 sing motion in any other directions. Now from the principle of 

 rectilinear axes of propagation (which results from reasoning given 

 in a paper " On a New Fundamental Equation in Hydrodyna- 

 mics," in the Cambridge Philosophical Transactions, vol. viii. 

 part 1, art. 7, and in a communication to the Philosophical 

 Magazine for December 1852, Prop. X.), it follows that this 



