Astronomer Royal on the Velocity of Sound. 4-97 



Here it is observable that no relation exists between the 

 points of no velocity and the points of maximum velocity. As 

 m, ^1, and A are arbitrary constants, we may even have 



mt^ — - = 0, 



in which case the points of no velocity are also points of maxi- 

 miim velocity. This is a manifest absurdity. No step, how- 

 ever, of the reasoning by which this result has been obtained 

 can be controverted. What then is the meaning of it ? Clearly 

 the analysis rejects the supposition of plane-waves, by giving 

 an integral which admits of no })hysical interpretation. Plane- 

 waves are thus shown to be physically impossible. 



Another way in which udx -{■ vdy + vodz becomes an exact 

 differential, is to suppose the velocity to be a function of the 

 distance from a fixed centre, that is, to suppose the waves to 

 be spherical. In this case an exact integral is not obtainable. 

 But it may be shown by an integral of the known approximate 

 equation 



dt^ ' dr^ ' 



that the analysis rejects this supposition also. The equation 

 is satisfied if r<p=y( ?•—«/); whence 



We may therefore have 



m . lit , 

 as= — sin — (?• — at): 

 r A ^ " 



and putting r — at = c, it follows that a given phase of the wave 

 is carried with the uniform velocity a, the condensation (s) 

 varying inversely as the distance ?•. This conclusion is gene- 

 rally adopted : but a very simple application of the principle 

 of constancy of mass will prove that it is false. For if c-j, o-g 

 be the condensations in spherical shells of indefinitely small 

 tliickiiess «, at corresponding parts of the same wave when at 

 different distances ?•,, Tj, the above-named principle requires 

 that 2t;-j0-,« should bo ecjual to 'Zirr^if^a, so that 7-'^^<r^ = 7-'^^(T,2, 

 and the condensation varies inversely as the square of the di- 

 stance. The inevitable conclusion from this reasoning is, that 

 the analysis rejects the supposition of spherical waves. 



There remains the supposition which I have adopted to 

 satisfy the condition that ndx + vdij-\-ixdz be an exact differen- 

 tial (r/\J/), viz. that \|/ is the product of a function of .r and j/, 



Pliil. Mag. S. 3. No. 218. Suppl. Vol. 32. 2 K 



