VOL. XXVI.] PHILOSOPHICAL TRANSACTIONS. 475 



HMN, hmn, will by no means be concentric circles to the sonorous body, but 

 curves of another kind, which yet must be similar to each other, and similarly 

 posited ; therefore, if in the Archbishop's hypothesis, the extreme wave alg, 

 touching the outmost confines of the terraqueous globe, be an hyperbola, all 

 the other intermediate waves hmn, hmn, must also be similar hyperbolas, and 

 similarly posited ; and though described from different vertices a, h, h, yet from 

 the same centre to the same axis, and under similar figures of the latus rectum 

 et transversum ; for whatever reason proves, on account of the simultaneous 

 appulse of sound to the points a, l, g, in the synchronal rays cha, cml, cng, 

 that ALG becomes a curve of such a kind, suppose a hyperbola ; the very same 

 reason will, on the same principles evince, that also, on account of the simul- 

 taneous appulse of sound to the points h, m, n, along the synchronal rays chn, 

 cmM, cnN, the wave hmn will become a curve of the same kind, viz. in this 

 case, a similar and similarly posited hyperbola, as is evident. And moreover it 

 is plain that the sonorous rays cha, cml, cng, should always intersect those 

 similar waves alg, hmn, hmn, perpendicularly, or at right angles, as is the case 

 in circular waves ; as Huygens demonstrates of the waves of light, in p. 44 of 

 the French edition of his Treatise on light. 



Therefore the investigation of the path, along which the sonorous rays, ac- 

 cording to the Archbishop's hypothesis, are propagated, is reduced to this 

 purely geometrical problem, viz. " To find the nature of those curves that inter- 

 sect perpendicularly any similar hyperbolas, and similarly described from the 

 same centre, and about the same axis." Let the similar hyperbolas alg, hmn, 

 hmn, fig. 1 6, and other innumerable intermediate hyperbolas, be similarly 

 posited, either above or below, having the same common centre o, and described 

 about the same axis oah, whose conjugate is os : through the point c draw the 

 curve cmML, or chng, intersecting all the given hyperbolas perpendicularly: 

 and through the given point c describe between the asymptotes oa, os, the 

 hyperbola cmML, of such a nature, that putting the ratio of the latus transver- 

 sum of the former hyperbolas al, hm, &c. to the latus rectum of the same, equal 

 to the ratio of t to r ; and supposing the powers of the ordinates lq, denomi- 

 nated from the exponent r, to be reciprocally proportional to the powers of the 

 abscissae oa from the centre ; that is, putting oa = x, and ol = y ; so that 

 yr r= x~' ; or drawing any other ordinates wi/, mi, so that the ratio of the dis- 

 tances oa, ox, from the centre, be reciprocally as the multiplicate of the ratio of 



the ordinates im, aL, as the fraction ^ is multiple of unity. This I say resolves 

 the question : for drawing the tangent lp of any hyperbola al, to a point where 

 it is intersected by the curve cml, as also slr the tangent of the hyperbola 

 cml to the same point, it is plain, from what I have shown in the demonstration 



3p2 



