The Rev. S. Earnshaw on the Triplicity of Sound. 187 



In the case of all ordinary sounds, h is indefinitely small com- 

 pared with Xj and the last term (as stated in art. 5) may be neg- 

 lected. But turning our attention to all possible sound-waves, 

 itthe last term may not be neglected, but will become more sen- 

 sible in proportion as X is less and less. This term leads to 

 results of great theoretical importance. For if we eliminate X 

 between equation (10) and the reduced equation for k (3), we 

 obtain 



l-.hk=Vv-v*, (11) 



which expresses the exact relation between k and v in the case 

 of every sound of the non-violent class as defined in art. 10. 



15. The symbol k is proportional to the number of vibrations 

 per second executed by any given particle, and, being so, may be 

 taken as a measure of the pitch of the note sounded. This being 

 agreed upon, the equation just found, being a quadratic in v, 

 shows us that there are two different velocities with which a given 

 musical note may be transmitted. The equation shows, moreover, 

 that the sum of these velocities is always equal to V ; and hence, 

 in the case of such musical notes as are not too high for audibility 

 by human ears, one of these velocities must be indefinitely small, 

 since the other differs (see art. 5) insensibly from V. And, 

 further, the length of the wave by which sound is rendered 

 audible to human ears is always large compared with h, the 

 distance between two neighbouring particles of air; but the 

 length of the second wave, by which the same note is transmitted, 

 is always extremely minute, never exceeding 2h; and conse- 

 quently this wave is too minute and feeble as to quantity of 

 momentum to affect such ears as ours ; and if audible at all, can 

 be so to none but the most minute creatures whose existence 

 has been revealed to us by the microscope. And if any doubt 

 should be entertained as to the existence of this minute order of 

 waves on account of their having never been perceived by the 

 experimentalist, let it be remembered that equation (11) is exact, 

 and that there is consequently no more ground in theory for 

 believing in the existence of those finite waves whose existence 

 is admitted, than of these extremely minute waves whose theo- 

 retical existence is now for the first time pointed out. Equa- 

 tion (11) indicates that one kind of wave has in theory as real 

 an existence as the other. I take it therefore as proved, that a 

 musical note of any pitch is transmissible with two different velo- 

 cities ; and that there are two waves for every note. 



16. Yet there is one essential difference between the two waves 

 corresponding to the same note, which is indicated by the fol- 

 lowing properties, viz. that the length of the ordinary wave in- 



02 



