364 Prof. Challis's Determination of the Velocity of Sound 



ance, particular forms of <p and /, which define a particular 

 kind of vibrations, it is shown that in all cases of small dis- 

 turbances the motion is composed of such vibrations. As the 

 vibrations are symmetrically disposed about an axis, I have 

 called them ray-vibrations. The number of the rays (the 

 equations defining them being linear), the directions of their 

 axes, and the values of m and A, may be assumed so as to 

 satisfy the conditions of given disturbances. 



This theory is not complete without obtaining a numerical 

 value of the velocity of propagation. It is unnecessary to re- 

 peat here in detail the mathematical reasoning by which I 

 succeeded in doing this in the Postscript to my communication 

 to the February Number of this Journal. It will suffice to say 

 that by making /=0 in the equation (8.), an equation results, 

 which is satisfied by an infinite number of values of r, such 

 that the difference between two consecutive values approaches 



continually, as r increases, to the limit ~~7^' These values of 



r correspond to positions of no condensation, since to the first 

 approximation we have 



The values of r which satisfy the equation -4-=0, are also 



unlimited in number, being intermediate to those which satisfy 

 y=0, and correspond to positions of no transverse velocity, 

 since the velocity transverse to the axis of the vibrations is 



<p -J-. Between two consecutive cylindrical surfaces of no 



velocity at an infinite distance from the axis, the transverse 

 vibrations are the same in kind as those which would take 

 place along the axis between two points of no velocity, sup- 

 posing two series of waves exactly equal were propagated 

 along that axis in opposite directions. The effect of the si- 

 multaneous propagation of two such series would be, to pro- 

 duce vibrations along the axis like the transverse vibrations 

 at an infinite distance ; and as the time of a vibration would 

 be the same in the two cases, it follows that 



—p. = -, and -^——. 



Hence, from what has already been shown, the velocity of 

 propagation is 



""s/ 



4 



1 + ^. 



