478 PHILOSOPHICAL TRANSACTIONS. [aNNO IJOQ, 



tangents, drawn from the centre of the earth through the extremities of the 

 vibrating fibre; and indeed the tremours of that fibre would not be propagated 

 in any other direction than in the lines t2, t3, t4, and other intermediate lines, 

 comprehended by the angle 2t4, and corresponding to each particle of the same 

 fibre; therefore the space without the said hyperbolas 298g, 476g, would have 

 no harmonical tremours, nor according to the Archbishop could the phonical 

 sphere be extended to the entire hemisphere of the earth ; therefore no one 

 single fibril of a sonorous body can ever really vibrate, but it must also affect, 

 and in like manner solicite to harmonical tremours other fibres, with which it 

 is connected, and between which it lies distended; which again must affect 

 others ; just as the tense string of a musical instrument evidently communicates 

 its tremours or vibrations to the wood to which it is fastened ; therefore on 

 {Striking the fibre of a sonorous body, the harmonic oscillations are transfused 

 into other bodies, with which it is either mediately or immediately connected ; 

 though they being always more and more weakened, and at length becoming 

 insensible, are farther and farther diffused over the superficies of the terrestrial 

 hemisphere, as appears on applying the ear to the ground, or hearing great 

 sounds at a considerable distance: therefore other sonorous hyperbolic rays pro- 

 ceed also from other places, by which the Archbishop's phonical sphere may be 

 sufficiently filled. 



To proceed to the second query, and give a more general solution of it ; sup- 

 pose any ray, either of light or sound, Nno, changed by its continual refrac- 

 tion into any curve; the question is, to find by what law the density of the 

 medium at different heights must be supposed to be varied, that supposing the 

 sign of refraction to be always proportional to the density of the refracting me- 

 dium, that ray may be formed into the curve sought. 



Let the axis of the curve nug, fig. l6, which the refracted ray describes, be 

 the right line co, in which taking any point c, describe with any radius cl the 

 quadrant of a circle LPp; and drawing any where a tangent nr, nr, to the 

 refracted ray, from c draw a radius parallel to the tangent, and meeting the 

 quadrant in p, and drawing pp parallel to the axis, let it meet the ordinate nq, 

 perpendicular to the axis in the point p; the curve, pfp, hence formed, expresses 

 by its ordinates FQ, fq, the densities of the medium, at different heights; for cp 

 being parallel to rn, the angle pcb will be equal to the angle which the refracted 

 ray NH forms with the perpendicular in the point n; and therefore bp, or p« 

 will always be the sine of refraction, taking cp for the radius ; therefore, sup- 

 posing this law of refraction, viz, that the sine of refraction is proportional to 

 the density of the medium, the same pa will express the density of the medium, 

 at the height a or n, a point erually high, through which the ray passes, a. e. d. 



