440 Mr. J. J. Walker on the Iris seen in Water. 



rays from any drop {d) makes with the line drawn in the direc- 

 tion of the sun's centre {s) -, let this pencil, after undergoing 

 reflexion at the point ((/) at which it meets the surface of the 

 water (which may be supposed to coincide with a tangent plane 

 to the surface of the terrestrial spheroid, at the foot of a ver- 

 tical, ec, let fall from the spectator's eye, e) enter the spectator's 

 eye : the locus of ff on the horizontal plane will evidently be a 

 line of red-tinted spots, forming part of the exterior band of the 

 horizontal iris. 



Let the vertical ec be produced beneath the horizontal plane 

 to €^, so that ce' = ec; then d^, if produced, will intersect the 

 line ece' at the fixed point e! : supposing, therefore, a line se'f 

 drawn through e' in the direction of the sun's centre, the line d^, 

 as it varies for successive drops, will generate a right cone about 

 e'f as axis, of which e^ will be the vertex. The locus of ^ will 

 therefore be the hyperbola in which this cone is intersected 

 by the horizontal plane. The centre and axes of the hyperbola 

 may perhaps be thus most easily found in terms of, h = ec= 

 height of spectator's eye above the horizontal plane, p the devia- 

 tion for red rays as above described, and a the altitude of the 

 sun's centre. Let a vertical plane through the sun's centre and 

 through the eye intersect the cone in the side 6/AV, and the 

 horizontal plane in the line cA ; A will plainly be the vertex of 

 the hyperbola, and cA its transverse axis ; its centre (o) in this 

 line will be found by drawing from e' in the vertical plane a line 

 e'o, making with the production of se' an angle (i) such that 



tan i . tan «=tan^p. 



The angle between the asymptotes is equal to the angle between 

 the two sides in which the cone is cut by a plane through the 

 vertex parallel to the horizontal plane ; and, calling this angle 

 2<j>, it is easy to see that 



cos <^. cos a = cos p. 

 Hence, for the speciesandpositionof the hyperbola, we have — 

 ratio of semiaxes 



= tan<6=if2li^±fl£e!(PZf!U!; . . . (1) 

 cosp 



distance of the centre from the foot of vertical let fall on the 

 horizontal plane from the spectator's eye 



n sec o 



distance of vertex of hyperbola from the same point 



=cA = hcot{p-a) (3) 



Since, for the same value of p, the rain-drops forming the cor- 



