Astronomer Royal on the Velocity of Sound. 495 



The results alluded to, viz. the non-divergence of the vibra- 

 tions, and a greater velocity of propagation than the vahie a, 

 are obtained by the part of the reasoning terminating, as 

 already specified, in page 280, before the introduction of the 

 new equation, which is first mentioned towards the bottom of 

 page 281. This equation in no way applies, unless the rea- 

 soning be carried beyond the first order of approximation. 



I admit that there is inconvenience in referring to a separate 

 investigation, contained in another work, and undertaken for 

 a different purpose ; and for this reason, as well as for another 

 which I shall mention hereafter, I will omit all consideration 

 of the equation against the introduction of which Mr. Airy 

 protests. If, however, the results derived from the approxi- 

 mate equations (3.) and (4'.) be liable to objection because 

 those equauons are approximate, it will be proper to investi- 

 gate and to use the corresponding exact equations, which 1 

 am prepared to do in the pages of this Journal. The course 

 of the argument will thus be clear, and equations (3.) and (4.) 

 will be employed together as "fairly and honestly" as Mr. 

 Airy can desire. 



I now maintain that I have proved, supposing b-, or e, to 

 have a value different from zero, that the waves are non-diver- 

 gent, and that the velocity of propagation is greater than a. 

 At the bottom of page 279 of my paper, the following equation 

 is obtained by reasoning which has not been objected to, 



Stt/ / e>:- ,\ . 

 <p = mcos — {z — at\/ H Y +^ J ' 



from which, since 



df df .d<p , / d<p 



it follows that 



u = m-r- cos — I z 

 ax A \ 



-^cos— (z — at \/ 1+ -o +c'] 

 d^ h\ V TT- / 



=_?!^/si„^(.-.^\/rrf +^) 



«,= __/^l+ ^sm-(^.-«/Vl+^-fcj. 



As/ is a function of .r and j/ only, these equations show that 

 the velocity and density at all points of any j'^ane perpendi- 



v—m , 



w 



