138 THE EEV. S. EAEXSHAW OX THE ]\IATHTAIATICAL THEOEY OF SOTXD. 
words, transmission-velocity is superimposed on particle-velocity, and in tMs sense 
transmission-velocity is everywhere the same, and equal to \/ fjj. A wave passes by 
every particle with this velocity, whatever be the particular and varying density of the 
medium where the particle is situated. 
17. Since a wave’s front cannot move faster than with the velocity -y/ if the gene- 
rating piston move faster than with this velocity, it will generate a bore ; and from this 
we infer that a bore always moves with a velocity greater than yj ; for wherever a bore 
may be situated at any ti7ne, we may suppose it to be just generated by a piston. If 
we write \/ yj for U, we find^=|M/g>=|7.-g>(,£=sy)o. Consequently if the piston press upon 
the resisting air with a pressure exceeding g atmospheres, a bore will be instantly fonned. 
I have defined a bore to be a tendency to discontinuity of pressure ; and it has been 
sho’wn that as a wave progresses such a tendency necessarily arises. As, however, dis- 
continuity of pressure is a physical impossibility, it is certain Nature has a way of avoid- 
ing its actual occurrence. To examine in what way she does this, let us suppose a discon- 
tinuity to have actually occurred at the point A, in a wave which is mo'sing foi'wards. 
Imagine a film of fluid at A forming a section at right angles to the tube. Then on 
the back of this film there is a certain pressure which is discontinuous with respect to 
the pressure on its front. To restore continuity of pressui’e, the film at A will rush for- 
Avard with a sudden increase of velocity, the pressure in the front of the film not being 
sufficient to preserve continuity of velocity. In so doing the film will play the part 
of a piston generating a bit of wave in front, and a small regressive waA’e behind. The 
result will be^ a prolongation of the wave’s front, thereby increasing the original length 
of the wave, and producing simultaneously a feeble regressive wave of a negative 
character. 
Now all this supposes the discontinuity to have actually occimred, which, as has been 
said, is a physical impossibility. For actual occurrence we must therefore substitute a 
tendency to occur, and modify the preceding reasoning thus : — 
Nature so contrives, that as the discontinuity is in its initial stage of beginning to 
take place, its actual occurrence is prevented by a (jradual (not sudden) prolongation of 
the weave’s front, and by the constant casting off, from its front in a retrogressNe dmec- 
tion, of a long continuous wave of a negative character, AAdiich Avill be of greater or less 
intensity according as the tendency to discontinuity is more or less intense in the 
original wave. 
The casting off of this long Avave Avill probably manifest itself audibly as a contmuous 
hiss or rushing sound. 
Hence a sound-Avave, from the moment that a tendency to discontinuity begins in its 
front, has the property of constantly prolonging itself in front, and by this means its 
front travels faster than at the rate \/ y>. Those sounds also aaIII travel most rapidly 
Avhose genesis Avas most violent ; and gentlest sounds travel AA'ith A'elocities not much 
differing from \/ yj. I should expect, therefore, that in circumstances AA’here the human 
voice can be heard at a sufficiently great distance, the command to fire a gun, if instantly 
