112 THE RAINEO\Y. 



of a transparent globe, tliey will be reflected from witliin so as to emerge, still 

 parallel to each other, at a point on the other side of the centre. The emergent 

 rayy will form a constant angle with the incident rays, and, entering the eye 

 of the observer standing with his back to the sun, will form the same angle with 

 a line supposed to be drawn from the sun through the eye. This line from the 

 sun through the eye being made an axis, and the above supposed reflected ray 

 being revolved around it, there will be traced out in the heavens a circle, from 

 every part of which, if rain-drops are present, there will come an amount of 

 light above that which is reflected from the surrounding cloud. 



This explanation satisfactorily determines the loais of the bow; but it fails 

 to account for its tints, or the extent of surface over which they are spread. It 

 would require that the arc should be white, and that it should be no broader 

 than the sun ; that is to say, that its brc^adth should be only about half a degree. 

 The actual breadth of the inner bow is, hoAvever, two degrees and a quarter; 

 and that of the outer three degrees and three quarters. Newton's discovery 

 furnished the necessary supplement to the theory. 



In fact, if the circumference PP'P" be a section 

 through the centre of a transparent globe, and IP a 

 ray of the sun falling on it in this plane, it is easy to 

 see that this ray. or portions of i^ will itndergo many 

 reflections within the globe, Avhile portions will succes- 

 sively emerge at the points in which reflection takes 

 place. There will first be some loss by external reflec- 

 tion in the direction PR. The portion which enters 

 the globe will be bent, by refraction, from the original 

 direction PK to the direction PP'. At P' a portion 

 will emerge in the direction PE, being bent from the 

 direction PQ as much as PP' Avas bent from PK. 

 The same thing occurs at P' , P"', and so on. Put j = the angle of incidence 

 (the angle made by the incident ray Avith the radius; — the angles of emer- 

 gence are all of this same value. Put /> for the angle of refraction. The figure 

 shows that all the angles of internal reflection have this value. Let <5 represent 

 the bending or deviation of the ray by refraction at each incidence or emer- 

 gence. Then <5rr:£ — p. And the amount of deflection of the successive re- 

 flected rays from the original direction being represented by D, D'' — -I^('")' 



and that of the successively emergent rays by J, J' D("^), Ave shall have 



(an entire circumference being denoted by 2-) 

 Deflection of PP'=o; deflection of PE=J=;2'r 



Deflection of P P"=D=2cJ+-— 2/> ; deflection of P"E'=J'=2o + -— 2/). 

 Deflection of P"P'"=D'=.5-f 2-— 4,o; deflection of P'''E''— J"— 2(J+2-— 4/>. 

 Deflection preceding m\\\ emergencerrD(^"'):^(J + 77«(- — 2^o) ; deflection m\\x 

 emcrgence=ziI(™j=2') + ;«(- — 2,o). 



If, for 0, Ave put its value ir^t — [>, we shall have— 

 J'— 2£ + -— 4,o. 

 J"— 2.' + 2-— 6,o. 

 J(m)=2:-f7«-— 2(7«-f l),o. 



The laAV of the formation of these expressions is obvious. The deflection 

 of each of the succi^gsively emergent rays is increased at each reflection within 

 the globe by the angular amount - — 2/^. 



Now, as all these values contain the angle i, it is obvious that the deflections 

 cannot be equal Avlien the incidences are unequal ; or, in other AA'ords, that the 

 (■inergent rays will usually diverge from each other. Moreover, the deflections 

 do not regularly increase and diminish Avith the incidence. 

 Putting the iiicidence=^0°, J'^ISO^', and J"-=3G0°, 

 Putting the int:idence^=90^, J'=16G°, and J"=24S°, ) i /• 



Putting the incidence^? 0% J'.^139^ and J"-^229^ j ^^'^"^'' *°^ ^^''^^®^- 



