484 Mr. J. J. Waterston on the Theory of Sound. 



fluid whose density is as the compression/^ he adds, in the scho- 

 lium, " But whether elastic fluids do really consist of particles so 

 repelling each other, is a physical question. We have here demon- 

 strated mathematically the property of fluids consisting of particles 

 of this kind, that hence philosophers may take occasion to discuss 

 tliat question." . .« . 



In Prop. 47 of Book 2, "Newton shows that " if pulses are pro- 

 pagated through a fluid, 'the .several particles of the fluid going 

 and returning with ' the shortest reciprocal motion are always 

 accelerated or retarded .Etccording to the law of the oscillating 

 pendulum." It is assu^i^d that the elastic force is proportional 

 to the density ; and in the direction of the pulse the fluid is sup- 

 posed to be divided into, physical lineolce, which are expansible 

 and contractile, and exhibit a force that resists compression in- 

 versely as their breadth. The mathematical reasoning defines 

 . the law by which the breadth of these lineolse must vary while 

 they go and return, — and hence the law of the diff"erence in the 

 breadth of two adjacent lineolse, and consequently the law of the 

 accelerative force operating on each corpuscle, which is thus 

 found to be the same as a body moving in a cycloid is subject 

 to under the influence of gravity. 



Newton's fundamental hypothesis is, that the particles of air 

 in the direction of the pulse are successively agitated with like 

 motions ; that both the dynamic condition and the static force 

 of repulsion, which is determined by the length of the line that 

 sepai'ates two adjacent particles (called a lineola), is transferred 

 onwards in the direction of the pulse from one particle to the 

 next adjacent in regular succession. 



The demonstration takes account of three orders of magni- 

 tudes : — 1, the breadth of a pulse (L) ; 2, the breadth of an oscil- 

 lation of a particle (3/) ; 3, the length of a lineola (X), each con- 

 sidered as infinitesimal with respect to the preceding. 



If the motion of a particle forward and backward in the line 

 21 corresponds to that of a cycloidal pendulum, i. e. if the rela- 

 tion between the accelerative force (acting in the line of motion), 

 the acquired velocity, and the time is the same in the line 21 as 

 in the complete cycloid, the force in this line must vary simply as 

 the distance (y) from its middle point. The value thus assigned 

 to the force implies that S, the difference between the lengths of 

 two adjacent lineolse, should vary also in this proportion. If a 

 semicircle is described on the diameter 21, y is the cosine of an 

 arc, of which x being the sine, we have h = y = dx; so that the 

 diff'erential of the hneola ought to be equal to the diff'erential of 

 the sine, and hence the absolute magnitude of the deviation of 

 the length of a lineola from its mean length ought to be pro- 

 portional to the sine. 



