Small Vibratory Motions of Elastic Fluids. 279 



because this equation, by reason of the unlimited number of 

 terms, may be made to belong to any curve possessing the re- 

 quired properties, by properly disposing of the values of m, m', 

 m", &c. The primary form, however, in all these curves, is that 

 given by the equation 



y = m\ sin — , 



and the motions they indicate, result from a composition of 

 motions indicated by this curve, as will be proved in a subse- 

 quent Article. It is this relation between y and s, which occurs 

 in Newton's famous forty-seventh Proposition. He jiroved that 

 the excursion of a particle may follow the law of a vibrating 

 pendulum, and in consequence that its velocity may vary as 

 the sine of a circular arc, representing the time from the be- 

 ginning of its motion, on the scale in which the whole circum- 

 ference represents the time of an excursion in going and returning. 

 It may here also be observed, that the reasoning- by which we 

 determined the velocity of propagation is fundamentally the 

 same as Newton's, since the determination depends on the 

 property -o = as, oji which Newton founds his reasoning in 

 Props. 48 and 49. His views appear to be quite correct as far 

 as they go: no one who objects to them can point out any 

 thing that is absolutely erroneous, and every one who defends 

 them, is willing to allow that they are incomplete. 



It has been shewn, that when two motions exactly alike 

 are propagated simultaneously in opposite directions, the state 

 of the particles at any instant is determined by the equations, 



■u = F {x - at) - F {x + at), 

 and as = F{x — at) + F{x + at), 

 and that there is a certain value I of x, for which 



F{1 - at) = F(l + at) 



