May 27, 1920] 



NATURE 



389 



of the water, which was only about a foot below the 

 level of our feet. 



The form of the bows is shown in Fig. i, which 

 is taken from a rough sketch made by the writer 

 from memory after returning home and before any 

 explanation of the phenomenon had suggested itself. 

 Here O is the observer in three positions, i., 11., and 

 III., and the bows seen from these positions are 

 numbered correspondingly. The twin bows seen in 

 position I. and the double curvatures at a in positions 

 II. and III. struck us as remarkable. As the observer 

 moved from the first position to the second, the nearer 

 branch of i. sank into the bank of the pond and the 



Edge of pond. 



Fig. 1. — O', O", O"', three positions of the observer's eye ; i., ii., iii., 

 the bows seen in these positions ; a, points of double curvature. 



further branch sprang out in a somewhat startling 

 manner until it attained the size roughly shown at 11. 

 The observer proceeding towards iii., the apex of the 

 bow, became lost in the distance. 



The bows were colourless except for a tinge of red 

 on the inside and of light of shorter wave-length on 

 the outside. They formed the limiting inner edge of 

 faint, diffuse luminosity extending over the general 

 surface of the pond, and within the bows was dark- 

 ness. In position ii. a ghost of an inner bow was 

 seen within the principal bow, roughly as shown in 

 the figure. No colour could be distinguished in this 

 inner bow. 



2.— S, source of light ; O, eye of observer; ax,a% angles of minimum 

 deviation for primary and secondary bows respectively. 



In the first position the surface of the water was 

 only about 5 ft. below the eye, and around the shadow 

 of the observer's head— on 'the fog as we thought at 

 the time, but on the water-surface as we now believe 

 —the diffuse light was brighter, forming an aureole 

 a few inches in breadth. When the head was 

 turned slightly, so that a ray of light from the lamp 

 to the water could pass close to the eye, the aureole 

 at the edge of the shadow near this eye became 

 brighter, suggesting that a ray reflected directly back 

 on itself was of considerable intensity. 



On subsequent nights we returned to the spot in 

 the hope of again seeing the phenomenon, but, doubt- 

 NO. 2639, VOL. 105] 



less owing to a change in the weather conditions, wei 

 saw nothing. We took measurements, however, and; 

 from these data Figs. 3 and 4 were calculated. ' 



The explanation of the phenomenon appears to be^ 

 as follows : — In certain weather conditions globules ofj 

 water are deposited from the fog upon the scum oni 

 the surface of the water, and the bows are formed inj 

 a way similar to those cast by the sun upon a bedewed^ 

 field — with this difference : that here the source ofj 

 light is near the observer. The bows, indeed, ar^ 

 rainbows. 



In order that a drop of water shall be so placed as] 



Fig. 3. — Primary bow, position i., as calculated, 



to return to the eye a maximum amount of light in 

 the manner that obtains in the rainbow, it is necessary 

 that the line joining the source to the eye shall sub- 

 tend an angle o at the drop, where o is the angle of 

 minimum deviation. The locus of such suitable posi- 

 tions for the drop is the surface of revolution of a 

 circle, of which the line joining the source to the 

 eve is a chord, about this chord (see Fig. 2). The 

 locus, then, is a toroidal figure like a flattened apple. 

 The locus for the secondary bow is such a toroidal 



Primary and secondary bows, position n., as calculated. 



figure within that for the primary bow. When the source 

 of light is at infinity, as in the case of the sun, these 

 figures become resolved into two co-axial conical 

 surfaces having the eye at their common apex. In 

 the present case the drops were confined to the plane 

 surface of the water, and the bows seen were plane 

 sections of these figures. 



In order to test the soundness of this explanation, 

 the writer has calculated the form of the bows from 

 the equation of the section of the toroidal figure made 

 by the water-surface, using as data the measurements 

 taken of the positions i., 11., and in., and giving a 

 the values 42° and 53° for the primary and secondary 



