1 38 Notices respecting New Books. 



of these functions allows of. But no person doubts that in alge- 

 braic functions, all the different values are connected together by the 

 law of continuity," {Misc.Taur. torn, i. p. 21.) Accordingly, no 

 discontinuous function can be exhibited to vieiv. Tiie inference to 

 be drawn with certainty from Lagrange's reasoning is, that if a num- 

 ber of particles, constituting a line of fluid, are in motion, the line 

 which bounds the ordinates erected at every point, proportional to 

 the velocities at a given instant, is not necessarily regular. It may 

 consist of portions of continuous curves, connected together at their 

 extremities, and be expressed analytically by a function, which pos- 

 sesses no distinctive property of discontinuity, but changes form 

 abruptly and in a manner always given by the data of the problem 

 to be solved. But if he had limited himself to this inference, and 

 not supposed the existence of a new order of functions, he could 

 not have determined the velocity of sound, and must have confessed 

 that the analytical theory had not succeeded in solving that problem. 

 For the demonstration he gives of it, rests altogether on the exist- 

 ence of discontinuous functions, such as they are above defined: and 

 herein it differs entirely from Newton's solution of the same problem, 

 which requires no new property of curves or functions, but deduces 

 the velocity directly from the constitution of the medium : — a me- 

 thod, which certainly at first sight appears the more natural. As, 

 however, we are sure that the velocity of the propagation of sound, 

 roust be a deduction from the principles on which the analytical in- 

 vestigation is founded, if no other mode of making the deduction 

 can be thought of, we must be content to take up with discon- 

 tinuous functions. No person can object to them who does not sup- 

 ply an equivalent, provided always they be considered in the pre- 

 sent state of analytic science, not as demonstrated to exist, but as 

 hypothetical, and like all hypotheses, established only by the extent 

 and success of their applications. It was necessary to premise so 

 much as this about discontinuous functions, in order to give a rea- 

 son why any one, who treats of the vibrations of an elastic medium, 

 has a right, if he can, to leave these functions out of consideration ; 

 and that the best possible argument for their non-existence is, to 

 show 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 principle, 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 infer- 

 ences drawn from such discussion, must also apply 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 



