Small Vibratory Motions of Elastic Fluids. 273 



consideration; and that the best possible argument for their non- 

 existence is, to shew how to do without them. 



In the dissertation that follows, I have reasoned as if all 

 functions were per se continuous, and setting out with this prin- 

 ciple, have discussed the integrals containing arbitrary functions, 

 prior to any supposition about the mode in which the fluid was 

 put in motion; considering that as the investigation which led 

 to these integrals was conducted without reference to any such 

 supposition, and as they are consequently applicable to every 

 point in motion, all inferences drawn from such discussion, must 

 also ajjply to every point in motion. This method of treating 

 the subject, dispenses with that of D'Alembert and Lagrange, 

 who consider the differential equation of the motion, to be 

 equivalent to an infinite number of equations of the same kind 

 as itself, each of which applies to a single point. The first 

 inference drawn from this manner of reasoning on the motions 

 in space of one dimension is, that every point is moving in such 

 a manner, as results either from a motion of propagation in a 

 single direction, or two simultaneous motions of propagation in 

 opposite directions. The velocity of the propagation is deter- 

 mined, and is, for air, the quantity commonly obtained by theory 

 for the velocity of sound. Again, it is shewn that the forms of 

 the functions are not entirely arbitrary, but limited by the 

 nature of the question to a certain species, the primary form of 

 which corresponds to the curve that occurs in Newton's reason- 

 ing, and by writers on the theory of vibrating chords called the 

 Taylorean Curve. As any number of these curves will simul- 

 taneously satisfy the partial differential equations, it is inferred 

 that the vibrations they indicate, may co-exist. If any portions 

 of these curves, or of the curves resulting from the combination 

 of any number of them, be joined together at their extremities, 

 and .so form an irregular line, every two consecutive ordinates 



Vol. III. Part I. Mm 



