Velocity of Sound. 129 



agreeably to what actually takes place in nature, and suppose 

 that the elastic force of the agitated cylinder is exerted while 

 it retains the whole of its absolute heat, the preceding formulge 

 (D) * will furnish this equation, 



P' \ / J dxJ 3 ' dx 



Take the fluxions making x only variable |, and divide by the 

 equal quantities § (dx-\-dz) and ^'dx ; then 



rfP ~ - i_ £1 _^ 



p idx + dz) 3 ' ^' ' dx^ * 



Now, P is the elastic force of the air in the tube at the dis- 

 tance x+z from the assumed point in the axis, and F+dP is 

 the like force of the air at the distance x+z+dx+dz ; wherefore 

 dP is the effective force urging the intervening cylinder towards 

 the assumed point: and as the mass moved is equal to § (dx+dz), 



♦ The formulae referred to make the cube of the pressure vary as the 

 fourth power of the density, which I consider to be the true law, though 

 Mr. Ivory has since renounced it as incorrect, without giving any ad- 

 missible reason ; but when he adopted this ratio, in the place from which 

 he now quotes it, he did so for an erroneous reason, as I have hinted in 

 the Edin. Phil. Jour, for January, ] 827. However, I do not think such 

 a ratio applicable to the investigation of the velocity of sound, especially 

 in the supposititious case of the tube before us. For though, in favourable 

 circumstances, sound be propagated in every open direction from the. 

 sonorous body, yet it does not appear that the air acts there exactly in its 

 fluid character. Because sound which first passes through the tube, and 

 then into the open air, does not proceed from the mouth of the tube, as 

 from a sonorous body, in every direction, which it would do if the particles 

 acted on each other with equal force in every direction. On the contrary, 

 sound, as is well known, diverges but in a small degi-ee after quitting a 

 long tube which merely conducts it ; and I rather doubt if it would di- 

 verge at all, were it not for the friction or resistance which the vibrating 

 particles suffer from their contact with air which is not in the direction of 

 the tube. From this we should be led to infer, that the particles of air 

 conveying sound through a narrow tube, es])ecially the ideal one free 

 from friction, only vibrate in the direction of the axis. If so, the elasticity 

 of air conducting sound through the tube should not be estimated accord- 

 ing to the above law, but more nearly as in the inverse ratio of the 

 squares of the variable longitudinal dimensions ; because, as I have 

 shown on a former occasion, the particles of air repel each other with 

 forces inversely as the squares of their distances. But we have already 

 seen that the actual case of the atmosphere is totally different from that 

 ofthetube.— H. M. 



t This is a curious injunction, more likely to embarrass and mislead 

 the reader than any thing else ; for the equation in hand does not involve 

 A' at all ; and, besides, Mr. Ivory, in the face of this strict precept, makes 

 both P and dz variable. — H. M. 



JULY— SEPT. 1828. K 



