LOOMING, OR HORIZONTAL REFRACTION. 155 



deviation of the place of the point s would be 32 seconds : Proportion of 

 the passage of the light through every 16 feet of the me- 

 dium producing a total deviation of a second; and if the ~ 

 change of the air's density were more or less than ~$ in the 

 space of a foot, the deviation would be as much move or less 

 than a second in each space of lG feet through which the 

 light passes. The curvature of the earth's surface becomes 

 a second in 102 feet ; consequently a change of density 

 amounting to -j^;, in a foot, or a change of temperature of 

 a degree in 6 or 7 feet, would be sufficient to produce a re- 

 fraction equivalent to the apparent depression of a distant 

 object arising from this cause, and to elevate the coasts of a 

 wide channel, so as to make them visible to each other. 

 This result ma}' also be more simply obtained from Simp- 

 son's investigations respecting atmospheric refraction, the 

 refractive density being inversely proportional to the distance 

 from the centre of the earth, when the temperature varies 

 1° in 6 or 7 feet ; for, as Dr. Young observes in his exten- 

 sive system of natural philosophy lately published, Vol, II. 

 Art. 46l, " If the refractive density of a medium varj r as a 

 given power of the distance from a certain central point, the 

 angular deviation of a ray of light will be, to the angle de- 

 scribed round the centre, as the exponent of the power to 

 unity." 



I am, Sir, 



Your very obedient servant, 



emeritus: 



Postscript. If it be required to determine the position of Formula for 

 X for the lowest ray that can cross the line £ ?, supposing t h e %hce of 

 it to be at any other distance from K, we must make the the image, 

 rectangle NUX=^NK«; as may be understood by con- 

 sidering that the fluxion of the tangent of P K X is in- 

 versely as N U, and the fluxion of K U is as U X. This 

 determination requires in general the solution of a biquadra- 

 tic equation ; but when K s is very small in proportion to 

 K N, U X will be very nearly ^ K «, or still more nearly 

 £K « + K s cub. -7 64 K N q. The point g thus found will 

 be the single point of the image as before : the length of the 

 path of the ray within the variable medium will in both cases be 

 M 2 half 



