THE EEV. S. EAENSHAW ON THE MATHEMATICAL THEOET OE SOUND. 139 
obeyed, and the rei^oTt of the gun, might be heard at a long distance in an inverse order ; 
i. e. first the report of the gun, and then the word “ fire*.” In a slight degree, therefore, 
the experimental velocity of sound will depend on its intensity, and the violence of its 
genesis. I consider this article as tending to account for the discrepancy between the 
calculated and observed velocities of sound (which most experimentalists have remarked 
and wondered at), when allowance is made (as will be done in a future part of this paper) 
for change of temperature. 
18. It seems reasonable to suppose that the audible character of a wave is in some 
way dependent on its tjrpe ; and consequently, if this be the case, the sound undergoes 
a perpetual modification as the distance of transmission increases. One modification of 
the sound-wave is, as we have seen, the formation of a bore in front ; but there is another 
which cannot but have some infiuence on its audible properties, as it corresponds to a 
remarkable change of type; and this takes place when the greater densities begin to 
overtake the less. 
Now when one degree of density overtakes another, the values of y corresponding to 
those two densities are equal ; and hence at the time t the equation 
y=Y+(y^+UX<-T) (12.) 
will give two equal values of y for two consecutive values of T. Hence differentiating it 
with regard to T, remembering that t is constant, or the same for both, as is also y, we have 
o=u-(y):+u)+(«-T)^, 
or 
(13-) 
dT 
The right-hand member of this equation is of course a continuous expression, and there- 
fore its least or minimum value will be the value of t when the modification of type, of 
which we are speaking, begins to take place; and because of the continuity of (13.), 
this modification once begun will gradually spread itself over the fore-part of the wave. 
Now # will be a minimum when 
From this equation we may find T, the time of genesis of that part of the wave where 
this modification begins. Then (13.) will give t, the actual time when the modification 
begins ; and (12.) will give the place in the tube where it begins. 
19. It is perhaps impossible to say what is the audible characteristic corresponding to 
the wave-modification just investigated ; but whatever it be, we perceive from (13.) that 
those sound-waves soonest begin to be affected by it for which ^ is largest ; ^. e. those 
* See Supplement to Appendix of Paeey’s Voyage in 1819-20, Art. “ Abstract of Experiments to deter- 
mine the Velocity of Sound.” 
