SB 



Astronomical and Nautical Collections. 349 



Let S be the centre of the earth, and ABC the ray of light, of 

 which AG and BG are tangents in A and B, meeting in G ; and 

 let SD, SQ, be perpendicular to the tangents ; join SA, SB, and 

 let AE, perpendicular to SA meet SD in E. Let the point A be 

 supiposed fixed, while the place of B is variable ; and call SA d=. a, 



= 6, SE = /, (= — ), SB = x, rfthe density at A, andy the 

 b J 



density in B. 



Now the curvature of the ray depends on the attractive force of 

 refraction . . which is always directed towards the greater density, 

 that is, towards the centre of the earth , . ; consequently the curve 

 is of the nature of a trajectory produced by a centripetal force. 



But the velocity of the light in A is to its velocity in B, (by the 

 Lemma) as s/ {\ + d) to aJ {I + t/) : consequently SQ : SD = 

 ^{l + d) : ^{l + y) (Cor. 1, Prop. 1, Book 1. Princ. Math.); 



whence SQ == V(l±^^ b, and BQ = V ^ - ^-^ b'. 'When 



va+2/) 1+2/ 



therefore the point B becomes infinitely distant, and y may be con- 

 sidered as vanishing, the perpendicular to the tangent, now become 

 an asymptote, will be // (1 ■^ d) b. We may suppose PH to be 

 that asymptote, meeting the tangents AG, BG, in F and H, and 

 SP to be perpendicular to it. 



Suppose now the tangent BG to be removed to a new place, bg, very 

 near the former, and let bg meet the perpendicular SQ in q. The 

 nascent angle gBG will then be [as] the fluxion of the angle FGH, 

 or FHG; that is, when x increases, it will be the elementary increment 

 of the angle FGH, and the decrement of FHG, because the angle 

 at F is given : and, for the radius =r 1, the arc proportional to the 



nascent angle QBq is —~^. But Qq is [as] the fluxion of SQ, or of 



Vl±^6, which is I3/V ^ + ^ b = «"y^V(l+^ b, since 

 1+2/ ^ {\ + y)i 2cx*(l +7/)l 



y ~ — — ^ , [that is, allowing for the decrease of gravity in the 

 cxx 



proportion of a* to x', and taking c for the modulus of the atmd* 



