68 



OF dov^jy. 



c c«: (1 eb f tSR g ab ti bb b c 



i 



-^^©.fe^o-sXfis: 



,(ytot|d^t^ o^ 



ScalesofC. 256 288 307 320341 



Equal tem- 

 perament. 

 Progressive X 

 temperaments./*^® 270 28? 3o3 321 341 360 383 405 427 455 481 512, 



384 409 427 451 480 512. 

 [•256 271287 304 323 342 36S 384 406 431 456 483 512. 



400. Theorem. All minute impulses are 

 conveyed through a homogeneous eUislic 

 medium with a uniform velocity, equal to 

 that which a heavy body would acquire by 

 falling through half the height of the me- 

 dium causing the pressm-e. 



If a moveable point be urged through a small space by 

 the difference of two forces, varying inversely as its distance 

 from two equidistant fixed points, in the same right line, 

 the times of describing that space will be ultimately equal, 

 whatever be its magnitude. For, calling the distance of 

 each point a, and the space to be described x, the forces 



TrviU be and , and their difTerence _ 



a—x a+x aa — xx 



,which 



IS to _ as nx to a ; but smce x is evanescent, this 



a a 



ratio becomes that of 2x to a, and the force varies as the 

 space to be described, consequently the times are equal. If 

 therefore all the particles of an elastic medium contiguous 

 to any plane, be agitated at the same time by a motion 

 varying according to any law, they will communicate a 

 motion to the particles on each side, and this motion will 

 be propagated in each direction with a uniform velocity, 

 and so that each particle shall observe the same law in its 

 motion. For, as in the collisions of elastic balls (344), 

 each ball communicates its whole motion to the next, and 

 then remains at rest, so each particle of the medium will 

 communicate its motion to the next in order; the common 

 centre of inertia of two neighbouring particles supplying 

 the place of a fixed point ; and the retrograde motions will 

 also be similarly communicated by the expansive force and 

 pressure of the medium ; and since the magnitude of the 

 motion, while it is considered as evanescent, -does not affect 

 the time of its communication from one particle to the next, 

 the velocity will not be affected by this magnitude, and the 

 whole successive motions will be transferred to the neigh- 

 bouring panicles in their original order and proportion. 



For computing the velocity, it is convenient to assume Sf 

 certain law for the motion of each particle, and it is 

 simplest to suppose it moving according 10 the law of the 

 cycloidal pendulum. Let AB 

 be the minute space described 

 by the particle A, in one semi- 

 vibration, while the undulation 

 is transmitted through AC- 

 DA, and let DE be a figiue 

 of sines, of which DA is the 

 half basis ; then if EF flow 

 uniformly with the-time, that 

 is, if it increase with the velocity of the undulation, the 

 versed sine FG will be in a constant ratio to (he motion of 

 A (259) ; the velocity of A will be as the fluxion of the 

 space, or of FG, that i?, as the conjugate ordinate HI 

 (142); the fo'ce will be as the fluxion of the velocity, or 

 as FG ; and the force being as the change of density, or as 

 its fluxion, the density, or rather the excess above the na.- 

 tural density, will be again as HI, and the fluent of'tfae pro- 

 duct of HI into the fluxion of the base, will giva the whole 

 excess of density in DA, which will therefore be represented 

 by the figure DAK (190). But when A arrives at B, ihe be- 

 ginning of, the undulation reaches C, and the whole fluid 

 which oci;upied A is condensed into BC, so that its meaa 

 density is increased in the ratio of AC to BC, and AB re- 

 presents the excess above the natural density; therefore 

 let the rectangle DLMA be to DAK, or DKq (203), as BC 

 to AB, or ultimately as AC or DA to AH ; that is, let 



DA.DL : DKq : : DA : AB, or DL: 



-DKq 



AB 



then DL will 



represent the natural density, while the ordinates HI every 

 where represent its increase. Let NA be the evanescent 

 length of the particle A, then the force actuating it will be 

 as the difference of the densities at its extremities, or as NO, 

 which is equal to NA (l4l) ; therefore the force impelling 

 A, is to the whole elasticity, as NA to DL. Now if h 

 be the height of a column of the fluid, equal in vreight ta 



