the Velocity of Soimd. 281 



fore required, as in the case of the integration of equation (3.)j 

 for deducing velocity of propagation. It may however be 

 argued, that as a particular value of (p was found, by which 

 the vibrations in the direction of ^ were defined, prior to any 

 consideration of the manner in which the fluid was put in 

 motion, so a particular value ofy exists (whether expressible 

 in finite terms or not), by which the condensation and velocity 

 in directions transverse to the axis of z are defined, and which 

 is equally independent of the arbitrary disturbance. Now the 

 form of equation (4.) clearly points to a supposition of a general 

 nature, by which that equation is converted into another con- 

 taining two variables, and consequently giving a particular 

 expression for/", viz. the supposition thaty'is a function of r 

 the distance from the axis ofz; according to which the con- 

 densation and transverse velocity are the same at the same 

 distance from the axis in all transverse directions. Thus di~ 

 rection of propagation is determined ab initio, in a manner not 

 depending on the particular disturbance, but which is common 

 to all disturbances; for plainly the initial direction of propaga- 

 tion is the axis about which the condensation is symmetrically 

 disposed. The above supposition respecting^' converts equa- 

 tion (4.) into the following, 



f +i;+*c/-=o, (6.) 



the integral of which in a series is, 



/=l-e;2+ j2^2- i2;2^2 + Scc-» • • • (7.) 



assuming that y=l, and y- =0, when r = 0. It is easily 



shown from this result, that there are an unlimited number 

 of possible values of/' for whichy vanishes. But we have no 

 right to conclude that equation (7.) gives the expression for/" 

 that we are seeking for, unless at some distance from the axis 



y and -J- vanish together ; that is, unless at some distance there 



be neither condensation nor variation of condensation, for 

 otherwise there will be transverse propagation. Now equation 

 (7.) does not satisfy this condition, as would appear by tracing 

 the curve which it represents. To meet this difficulty, re- 

 course must be had to the exact equation corresponding to the 

 approximate equation (4.). That equation I have obtained in 

 my paper on Luminous Rays (Cambridge Philosophical Trans- 

 actions, vol. viii, part 3, p. 368), to which, as the reasoning is 



