110 



EFFECTS OF EEFRACTION. 



angle of incidence to the co-secant of the angle of refraction is constant. As 

 the co-secants of angles are inversely as the sines of the same angles, the law 

 may be more conveniently expressed by saying that, in the circumstances sup- 

 posed, tlie sines of the angles mentioned arc in a constant ratio. It was in this 

 form that the law was first published by Descartes, eleven years after the death 

 of Snellius. It is, therefore, frequently referi'ed to as the law of Descartes. 



It may be proper to mention that, previously to the discovery of this impor- 

 tant law by Snellius, it had been remarked by the illustrious Kepler that for 

 incidences below thirty degrees a ratio almost constant exists between the angles 

 of incidence and of refraction themselves. This is true because for small angles 

 the increments of the arc and of the sine are nearly proportional. But when the 

 incidence is moderately large, the divergency of the two ratios becomes very 

 wide. 



An examination of the figure given above will show that the refraction of a 

 plane surface produces no distortion in lines which are at right angles to the 

 surface, but only diminishes or increases their apparent length according as the 

 medium in which the object is situated is denser or rarer than that on the side 

 of the observer. Thus the line ED is reduced to the apparent length EB. 

 The amount of this reduction increases with the obliquity of the visual ray, for 

 the ratio of CD to CB, which is constant, is always less (except when the 

 incidence is perpendicular) than the ratio of ED to EB, and the divergency 

 of these ratios is always increasing. It follows that the apparent depth of a 

 fluid is always less than the real depLh, and that the illusion is more striking in 

 proportion as the point obi^erved is more remote from that immediately beneath 

 the eye. Thus the horizontal bottom of a cistern or pool of uniform depth 



presents a curved appearance like 

 that here represented. If MN be 

 the surfacg and KL the horizontal 

 plane at the bottom of a sheet of 

 water, the eye being placed at the 

 point P above it, this plane will 

 present a conchoidal appearance 

 like that of the curve D 'E 'A". 

 ^J'he position of the points of the 

 bottom which, to an eye situated at P, appear iu the directions PD, PE, &c , 

 may be found by a simple geometrical construction. Drawing the perpendicular 

 PAA', divide the depth AA' at the point A.", so that AA' shall be to AA" 

 in the ratio of m to 1 : — n being the index of refraction. Through A" draw 

 VW parallel to the surface. Produce PD, PE, &c., until they intersect the 

 bottom at CI and II, and with the radii DE and EH describe the circular arcs 

 GG' and HIP. Through G" and II" where DG, EH, intersect VW, draw per- 

 pendiL'ulars to the bottom, intersecting the arcs in G' and IP. Join DG' and 

 EIP. The points D' and E', where the joining lines intersect the bottom, are 

 the points which will be seen from P in the directions PD, PE, and the ap- 

 parent positions of those points will be at D" and E", where the visual rays 

 PD and PE produced meet perpendiculars drawn from D', E', to the surface. 

 Any number of points being thus found, the curve drawn through thcin all 

 will show the appearance of the level bottom M'N' as it is seen from a point 

 above the surface as P. This curve is a conchoid, whose polar equation is 



r=2)i>cccp + seccr' ; 



VI^ 



in which j) is put for PA, q for AA', n for the index of refraction, <p for the 

 angle EPA, and <p' for EE'E". 



• It is apparent from the foregoing that all lines seen through a single plane 

 refracting surface, unless they are perpendicular to the surface iiself, are more 



