VOL. XXXII.J PHILOSOPHICAL TRANSACTIONS. 625 



surface. Suppose these rays to pass in the lines bf, bh ; the former of which 

 rays shall have had one fit more of easy transmission, and the latter one fit less, 

 than the rays that pass from b to c. Now both these rays when they go out 

 of the drop, will proceed by the refraction of the water in the lines pi, hk, 

 which will be inclined almost equally to the rays incident on the drop that 

 come from the sun, but the angles of their inclination will be less than the 

 angle in which the rays emerging in the line cd are inclined to those incident 

 rays. And after the same manner, rays scattered from the point b, at a certain 

 distance without these, will emerge out of the drop, while the intermediate 

 rays are intercepted; and these emergent rays will be inclined to the rays inci- 

 dent on the drop, in angles still less than the angles in which the rays fi and 

 HK are inclined to them ; and without these rays will emerge other rays, that 

 shall be inclined to the incident rays in angles yet less. Now by this means 

 will be formed of every kind of rays, besides the principal arch which goes to 

 the formation of the rainbow, other arches, within every one of the principal, 

 of the same colour, though much more faint; and this for divers successions, 

 as long as these weak lights, which in every arch grow more and more obscure, 

 shall continue visible. Now as the arches produced by each colour will be 

 variously mixed together, the diversity of colours observed by Dr. Langwith 

 may well arise from them. 



The precise distances between the principal arch of each respective colour, and 

 these fainter correspondent arches, depend on the magnitude of the drops of rain. 

 In particular, the smallest drops will make the secondary arches of each species 

 at the greatest distance from their respective principal, and from each other. 

 Whence, as the drops of rain increase in falling, these arches near the horizon, 

 by their great nearness to their respective principal arches, become invisible. 



Of refracting circles in two porisms. — I shall here set down two propositions, 

 which I have formerly considered, relating to this subject. For the greater 

 brevity I shall deliver them under the form of porisms; as, in my opinion, the 

 ancients called all propositions treated by analysis only. 



Prop, l . In a given refracting Circle, whose refracting Poiver is given, 

 the Ray is given in Position, which, passing Parallel to a given Diameter of 

 the Circle, is refracted by that Circle to a Point given in its Circumference. 

 — Let ABCD, fig. '2, be the given circle, the given diameter ac, and given 

 point g; and let the ray ef, parallel to ac, be refracted to g. I say ep is given 

 in position. 



Produce ep to h, and draw the diameter fi, drawing also ikh, ig. Then 

 is hfi the angle of incidence, and gfi the refracted angle; so that ih being 

 perpendicular to fh, and ig perpendicular to fg, ih is to ig as the sine of the 



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