36 REPORT—1857. ~ 
bow, a constant radius, while according to that of Mr. Airy, it ought to be variable, a 
matter which entirely agrees with observations made. I have added to my memoir 
a drawing, which represents, according to these two theories, the variations which 
the rainbow’s fringes of interference undergo according to the different diameters of 
the drops, from two millimetres to two hundredths of a millimetre, ard for the two 
extreme colours of the solar spectrum, the red next the ray C, and the violet next the 
ray Hi. 
The simply punctuated curves of this drawing give the deviation of the maxima 
and minima red and violct of different orders for a given diameter of the drops, the 
ahscissx of these curves representing the deviations of the different points of each 
fringe ; and the ordinates, the proximate intensities of these points in the theory of 
Young. The curves with a full and coloured line represent the real action of the 
phenomenon; they only give the variations of the first two fringes, red as well as 
violet, the numerical results of Mr. Airy having only furnished me with means of 
cyphering a greater number. But these first two fringes are by far the most import- 
ant, and are more than sufficient to fix the real theory of the rainbow, and of all 
the variations which it undergoes in its breadth, ray, and the shade of its colours ; as 
also the theory of supernumerary bows, the white rainbow, the crowns opposite 
the sun, which are nothing else but supernumerary bows, as I demonstrate; and 
finally, to explain the absence of the coloured rainbow in fogs and rainless clouds. 
The relative intensities of the different points of these two fringes are not indicated 
in an arbitrary manner by the ordinates of the curves which represent them. I have 
determined them as faithfully as I could, making use of the drawings and numbers 
published by Mr. Airy in his learned “ Memoir on the Intensity of Light in the 
vicinity of a Caustic.” 
The columns of figures contained in my Table give a facility in passing from one 
system to another, by means of a formula which I give in my memoir, and from 
proportional numbers taken from the memoir of Mr. Airy. The curves may be 
multiplied at will by choosing other measures of diameter than those I have given ; 
I have traced a sufficient number of them for the requirements of my thesis. Here 
are the principal results drawn from them. 
Instead of a deviation of 42° 16! for the red rays, and 40° 29! for the violet, which the 
first maximum should have presented, according to the theory of Descartes and that 
of Dr. Young, whatever may be the diameter of the drops, we find for a diameter of 
Dev. 2 1st Maximum red . 42° 3’ | Ist Maximum violet ‘ - 40° 20’ 
Dey. 1 a 5s pte bb! - as r . 40° 14’ 
Dev. 5 33 bs 0400043) # P 140 Se, 
Dev. 2 i» 5 aie Ady! = a : . 39° 47’ 
Dey. 0°1 53 spt 402 240 a pF , 2 1oOneee: 
Dev. 0°05 a peer, oan 4A. i *. 2 4 ieee 
Dev. 0°02 3 ea WaTeea 1% > 3 f Pay pee bit ss 
The distance of the first maximum to the second is, in drops having 
2 for the red rays . -  «, \ 32°\| For'the violet rays. | 24’ 
1 Fy if es2 ie - ° 53’ ae AS Ph S ° ° 38' 
(SO ee RRR Beal. Y me a eC 58" 
LE OE OAS Se Bd ARR eg oe oe ap in Ss So 
Ol ” ” ae © e . 4° 6' ” ” 23) = ° . 2° 50! ’ 
0°04 ,, “ ae ies ; e707 25! > » © |) 
The progressive diminution of the diameter of the drops causes a continually in- 
creasing augmentation in the breadth of the first fringe and all the colours, and ends - 
by their being superposed; such is the cause of the disappearance of the coloured 
rainbow in the clouds and fogs, and the origin of the white rainbow which after- 
wards disappears. ; 
When the diameter of the globules, of which the fogs are formed, become less than 
0:02 the millimetre, the same Table shows that the conditions favourable to the for- 
mation of the white rainbow, is, that the globules of the fogs have a diameter always 
between 0°1 and 0°02 of a millimetre. Moreover, if the diameter does not exceed 
0°05 of a millimetre, the colours of all the fringes which follow the first, present 
themselves in a bow, from that in which they are produced under the latter, and 
