\!A 



LOOMING, OR HORIZONTAL REFRACTION. 



Path of 

 of light 



the ray 



Apparent 

 place of the 

 object. 



Double. 



An inverted 

 object only. 



the variable strata from the medium, its motion will be simi- 

 lar to that of a jet or a projectile rising in any direction from 

 the bottom of a reservoir with the velocity due to its height. 



Let A 13 (Pl.V. Fig. ] .) be the imaginary height ; if we de- 

 scribe the semicircle AC B, a ray of light tntering.atB, in the 

 direction B C, will describe the parabolic path B D E, BE 

 being four times FC; or if the circle B G II be twice as 

 great in diameter as A B, F D will be equal to G I. And 

 if several rays, passing from a point K, (Fig. 2.) enter the 

 variable medium at the lower surface L M ; making the se- 

 micircle K N equal to B G H, the distances K O, K P, 

 will be equal to 2 Q L + 2 11 S, and 2 Q T + 2 U X, re- 

 spectively. 



Now the distance K P must be a minimum, when the 

 fluxions of QT and UX are equal; that is, when KV 

 s= X Z, (Fig. 3.) T Y being perpendicular to KT, and 

 X Z parallel to K N. Make Ka = KQ, and describe the 

 semicircles K 3, a. 3, Kg being half of K N ; draw K y 

 perpendicular, and y </ v parallel to KN ; then S" X, parallel 

 to Q T, will determine the position of the point X fo as to 

 fulfil this condition. It is obvious that when K Q is very 

 small in proportion to KN, «T X will coincide with Q T, 

 and X will be in the intersection of the circle K N with the 

 surface. 



Consequently to an eye placed-at K, (Fig. A.) the object 

 s will be seen in the direction K X, and the object £ in the 

 directions K »i and K d ; so that there will be two elevated 

 images of the line « £, the one erect, and the other in- 

 verted. 



If the variable medium be only thick enough to admit 

 the passage of the rays below K X, there will be no di- 

 rect image, but an inverted one only. The inverted image 

 will in general be nearly of the natural dimensions, although 

 a little contracted ; the case being nearly similar to a very 

 oblique internal reflection. The points K and ? may be 

 considered as conjugate foci, with respect to the refraction 

 of the variable medium. 



In the supposed case of the variation of a degree for each 

 foot of air, K N being 34- million feet, if K Q be 1 inch, 



1i X will be -3 i~i feet, and 



K g 700 yards. The angular 

 deviation 



