OF OPTICAL INSTRUMENTS. 



81 



Since the densities are supposed to be 

 equal at equal distances from the centre 

 the radius must always be perpendicular to 

 the direction of the refracting surface, and 

 t\ro perpendiculars falling on the direction of 

 the ray in any two points infinitely near 

 each other, will be the sines of incidence 

 and refraction for the intervening surface : this perpen- 

 dicular will therefore always vary inversely as the re- 

 fractive density ; and if the density be as the power q of 



the distance x, the perpendicular u will be as 

 Now v' {xx — uu) : « : : — l' : 



v^ax 



•/ [xx — uu) 

 which is the increment of the arc described by the radius 



accompany mg the ray, and is the fluxion 



x,,/ [xx — uu) 



of a similar arc of which the radius is I. But the fluxion 



, . b 



of the arc in the same circle ot which — u is the sine, that 



X 



IS, of the arc corresponding to the inclination of the ray to 



the radius, n bt ) . ; and since k= 



V X XX / v' (xx — uu) 



—q . —qux 



- = — q — , and the fhixion of this 



X XX 



luv 



arc becomes —(g-fi; ■ , which is to the 



x.^(a:x—uu) 



fluxion of the arc described as (q+l) to 1 : therefore the 

 finite increments of these arcs are in the same proportion. 

 But the difference between the angle described, and the 

 change of the inclination to the radius, is the angle of de- 

 viation, which is therefore to the angle described as g to 

 1, and to the change of inclination as q to q + l. 



Scholium. It is found that the circumstances of at- 

 mospherical refraction agree nearly with such a constitu- 

 tion of the medium, supposing gzi — \; but from parti- 

 cular circumstances which take place near the earth's 

 surface, the terrestrial refraction, instead of being l of the 

 arc intervening between two places, is seldom more than 

 ^. If any two values of x are given, the ratio of the cor- 

 responding values of » is also given, and the ratio of the 

 sines of inclination to the radius will be constant for any 

 rays passing through the same depth of the medium ; and 

 the angle of inclination being determined, the angle of 

 deviation will be to this angle in the ratio of g to j-l-l. 

 For the whole height of the atmosphere, the logarithm of 

 the ratio of the sines of inclination is .ooo"300, and q : q+ 

 1 : : 1 : 6 ; so that if we deduct this logarithm from the lo- 

 garithmic sine of the apparent zenith distance, we shall find 

 an angle, which differs from the zenith distance by six 

 VOL. II. 



times the refraction, at a mean height of the barometer 

 and thermometer. At the horizon the refraction is 33' ; 

 at the altitude 45°, by". 



462. Definition. The rainbow is pro- 

 duced by a combination of refractions and 

 reflections, which cause the sun's rays of dif- 

 ferent colours, to be transmitted most co- 

 piously to the spectator, from the spherical 

 drops of rain or dew, under different angles 

 of incidence. In the interior rainbow the 

 rays are once reflected at the posterior sur- 

 face of the drop ; in the exterior they are 

 twice reflected. 



463, Theorem. In the interior rainbow, 

 the tangent of the angle of incidence is 

 twice, in the exterior three times, that of the 

 angle of refraction. 



When parallel rays fall on a sphere, and after refractioa 

 at their entrance, are reflected again from the posterior 

 concave surface, and refracted a second time in their pat- 

 sage out of the sphere, the ray in the direction of the axis 

 emerges in the same line, but the lateral rays deviate more 

 and more from their former direction, till, at a certain dis- 

 tance from the axis, the deviation is again lessened ; and, as 

 in other maximums and minimums, the change of the 

 deviation being slowest when it becomes a maximum, the 

 light being most dense where this change is smallest, the 

 conical surface formed by the rays most inclined to the axis,- 

 determines the direction of ihe strongest light ; we must 

 therefore compute the magnitude of the greatest deviation. 



Now Z.ABC— "zABD (405) and ABD A X, B 



nEFD— BEFnDEF— BEF, there- 

 fore the difference between the angles 

 of refraction and of deviation at E, 

 must be a maximum, and their flux- 

 ions must be equal, therefore the fluxions of their sum 

 DEB must be double that of DEF. Now, when the sinea 

 of two arcs are in a constant ratio, the fluxions of the sines, 

 are in the same ratio as the sines themselves : and the 

 fluxion of the sine is to that of the arc, as the cosine to the 

 radius (142), or as the sine to the tangent (I2l), therefore 

 the fluxions of the two arcs are as the tangents. Hence the 

 tangent of DEB must be twice the tmgent of DEF. Let 

 their sines be rxandx; then their cosines will be •/ (l — r-x') 



and \/ (1— x'l, and their tangents and — : 



M 



