OPTICS.

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that next the perpendicular IGivill be red, as being least refract-ed ; and the rest in order ubuve it, viz. orange, yellow, green,blue, indigo, violet. Now it is found by computation, that thegreatest angle SEO, (tig. 4',) or EOP, because OP parallel toSE, under whu n the most refrangible lavs can come to the eyeof the spectator at O, is 40” i T ; and that the greatest angle FOP,under .wiib.n the least refrangible rays come to ttie eye at O, is42° 2': and so all the particles of water, withm the difference ofthose two angles, EF, will exhibit severally the various colours ofthe prism, and constitute the interior rainbow in trie cloud. Ifthe beam of light go not out of the drop, at G, but is reflected asecond time, to H ; (tig. 40 and 41,) and is there refracted in diedirection IIS, making the angle SYA with the incident ray AN,it will paint on the part H the several colours of light ; but in aninverse ore.'.:” >o the former, and more faint, by reason of the rayslost by the second reflection. It has been found also, that theleast angle SGO, or GOP, under which the least refrangible rayscan come to the eye at O, after two reflections and two refractionsis 50° 57'; and the least angle HOP, under which the most re-frangible rays can come to the eye in this case, is 54° 7': whenceall the colours of the exterior rainbow, will be formed in the dropsfrom G to II; which is the breadth of the bow, viz. 3° in';whereas the breadth of the former, or interior bow, viz. F.F, isbut 1° 45'; and the distance between the bows, viz. FG, is 8° 55'.And such would be the measure of the bows, were the sun but apoint; but since his body subtends an angle of half a degree, it isevident, by so much will each bow be increased, and their dis-tance diminished. To apprehend rightly the different affectionsol the rainbow, we must attend to the following particulars: 1.That though the rainbow be occasioned by the refracted and re-flected light of the sun falling on the drops of rain, yet neither ofthem is produced by any rays falling on any' part of the drop in-differently, but by those only which fall on ihe surface of the dropBLQG, (iig. 40, 41, and 42,) in or about the point N, as the rayAN ; those which fall nearer to B, or farther towards L, beinguncertain in this production. 2. The internal how is producedby two refractions and one reflection. The tirst refraction is ofthe incident rays extremely near AN, by which they proceedfrom N to one common point or focus at F, from whence theyare reflected to G, and are there a second time refracted towardsB, and produce the various colours of the said bow. 3. There isa necessity that several rays should be refracted together to thepoint F, that being reflected together from thence to G they maythere go out parallel, and so come in quantity sufficient to excitethe sensation of colours in a strong and lively manner. Nowthose rays, and those only, which are incident on the globuleabout the point N, can do this, as will appear from what follows:for, 4. The point F makes the arch QF a maximum, or the dis-tance QF from the axis of the drop SQ is greater than any otherdistance from whence any other rays nearer to the axis, as SD,SE, or farther from it, as SH, SI, are reflected; because thosewhich are nearer after the first refraction tend to points in the axisproduced more remote than that to which the ray SN lends; andtherefore as their distance from the axis increases, so likewise willthe distances of their points of reflection QP, QO, till the ray be-comes SN ; after which the rays more remote from the axis, asSH, SI, are refracted towards the points X, Y, which are nearerand nearer to the axis; and this occasions the points of reflectionon the farthest side of the drop to decrease again from F towardsQ. 5. Hence it will necessarily happen, that some rays aboveand below the ray SN will fall upon the same point, as O or P,on the farthest side ; and for that reason they will be so reflectedfrom thence as to go out of the drop by refraction parallel to eachother. Thus let SE below, and SH above the ray SN, be re-fracted both to one point O ; from hence they will be reflected toM and L, and will there emerge parallel, it is true, but alone;being divested of their intermediate rays SN, which going to adifferent point F'will be reflected in a different direction to G,and emerge on one side, and not between those rays, as when theywere incident on the drop. All which is evident from the figure.<j. As this will be the case of all the rays which are not indefinitelynear to SN, it is plain, that being deprived of the intermediaterays, their density will be so far diminished, as to render them in-effectual for exciting the sensation of colours; and they are there-fore called inefficacious rays, in contradistinction to those which

enter the drop near SN, and which, having the same point F ofreflection, are not scattered like the others, but emerge togetherat G, so as to constitute a beam GR of the same density with t)ieincident beam SN, and therefore capable of exhibiting a vividappearance of colours, and for this reason are called efficaciouslays. We may now account for the following phenomena of thebows. 1. Each bow is variegated with all the prismatic colours.This is a necessary consequence of the different refrangibility ofthe rays refracted and reflected in drops of falling rain. Let A,(fig. 43,) he such a drop, SN a ray entering it at N, which is re-fracted to F, from whence reflected to G, where, as it emerges,it is refracted into all the several sorts of rays of which it is com-posed, viz. GR, the least refrangible or red-making ray, GO theorange, GY the yellow, GG the green, GB the blue, GI the in-digo, and GV the violet, or most refrangible ray. The truth ofthis may be easily proved by experiment, by suspending a glass-globe filled with water in the sun-shine, and viewing it in such aposition, that the rays SN will fall upon it, and emerge to the eyeat A, under the several angles from SFR to SFV ; which may "beeasily effected by letting the globe descend from A to G, by astring going over a pulley. 2. Hence the circular form, is account-ed for, and also the breadth of tire bow ; for that will be equal tothe angle ARG = RGV = 1° 45', where the ray, as here, emergesafter one reflection. These particulars are represented morecompletely in fig. 41, where BI D is the red circumference form-ed by the rotation of the ray OF, that can first come to the eyeat O ; and CEA is the violet arch formed by the least refrangibleray EO ; after which the rays are all refracted below the eye.And thus, by the intermediate rays and colours, the whole interiorbow is produced. 3. The appearance of two bows. This followsfrom hence, that after an efficacious ray of light, entering a dropof rain, has been twice reflected on the farthest side, as at G andH, it will emerge refracted into all its simple or constituent raysat G and H, upon the upper side of the drop, so as to make withthe incident ray an angle 54° 10', if that ray be the violet sort, ormost refrangible ; but if it be the red or least refrangible sort, thenthe said angle is but 50° 58'. Therefore, all those drops whichare so situated around the eye, that their most refrangible raysshall fall upon it, must with those rays make an angle with theline OP passing through the eye parallel to the sun’s rays, viz. thqangle HOP, equal to the angle SYO, equal to 54° 10'. Theserays, therefore, will everywhere exhibit a violet colour in the archKHL. For the same reason, those drops whose least refrangiblerays fall upon the eye at O, make the angle GOP = 50° 53';and so the ray OG revolving about the axis OP, will describe thecircular arch MGN, which will exhibit the deepest red; and allthe drops between G and H will paint the several other colouredperipheries, all which together will complete the exterior bow-.4. The greater breadth of the exterior bow. Thus, if from 54°10'we subduct 50° 58', we shall have 3° 12' = GH = the widthof the outer bow ; which, therefore, is almost twice as wide as theinterior bow. 5. The distance between the two bows, which is-thus determined; from the angle which the least refrangible ray inthe upper bow makes with the axis OP, viz. 50° .58', subtract theangle 42° 2' which the most refrangible rays make therewith in thelower bow, and the remainder 8° 56' = GOF is the arch of dis-tance between the bows. 6. The inverse order of the colours intlie two bows, 'l'his follows from the contrary parts of the dropon which the ray is incident, and from whence it emerges and isrefracted. Thus, when the rays enter the upper part of the drop,and emerge from the lower, it is evident the rays refracted in thiscase (viz. in the interior bow,) will have a situation quite the re-verse of those which enter on the lower part of the drop, and arerefracted from the upper, as in the exterior bow, whose colours areviolet, indigo, blue, green, yellow, orange, and red ; whilst thoseof the other are red, orange, yellow, green, blue, indigo, and vio-let ; counting from the upper part downwards in both. 7. Thefaintness of the exterior bow, in comparison of the interior one.This is the consequence of the rays being twice reflected within,the drops which form the outer bow. They who make the expe-riment in a dark chamber, may wonder when they observe howlarge a part of the beam (that enters the globule at N,) goes outat F, (fig- 40,) that there should be enough in the remaining partFG to exhibit the colours so strong and vivid in the first bow asthey appear ; but then, considering how much of this residual ray; ' IS