THE EAINBOW. Ho 



Neither of the last values is intermediate Ixitwcen the two precefling in th(- 

 same column. In both cases, therefore, there appears to be some point between 

 the extreme incidc-'nces, where the deflection is a minimum ; and it being the 

 law of maxima and minima that variations in their vicinity are insensii)lc, it 

 follows that near the incidences corresponding to those values the emergent rays 

 will be sensibly parallel. But when the general expression — 



J ('i»^ =2 £ + ?«-— 2 7»+l)/> 

 becomes a minimum*, the cosine of the incidence must have the value — 



^Vr "-' 



{?}i+lf — 1, 

 in which n denotes the index of refraction. 



This determines, therefore, the incidences at which the deflections are minima : 

 and hence, those at which the emergent rays are (to use the term employed b}' 

 Newton) effic.ucious . It will be seen that, when the index of refraction is given, 

 the value of cost will be affected only by the variable m, which is the number 

 of internal reflections. If this be made zero, cost will be infinite; in oth"r 

 words, when the rays are not reflected at all, they do not emerge efficacious. 



By putting w=l and m=^2 we shall obtain values corresponding to the de- 

 flections which produce what are called the inner and outer bows. From these 

 values we may deduce the apparent diameters of the arcs ; and the theoretic 

 results thus obtained are found to accord with actual measurements. By putting 

 m==2, 4, 5, &c., successively, we may obtain the loci of an infinite number of 

 additional bows; but after the second reflection, the light ceases to be intense 

 enough to produce an impression on the eye. 



Snrce, with a very slight alteration of : the rays cease to be efficacious, it i^ 

 evident that, if the sun were but a point, and the index n invariable, the bow 

 Avould be reduced to a simple line of light. But as every point of tlie sun will 

 produce its separate bow, the visible breadth, with n constant, would be that 

 of the sun itself — that is, about half a degree. Newton's experiments on dis- 

 persion, however, showed that the value of the index n sufficiently varies, in 

 passing from the red to the violet, to alter sensibly the angle of inciclence cor- 

 responding to the efficacious rays of the several colors, and sufficient, accord- 

 ingly, to alter the amount of deflection which those several rays undergo before 

 reaching the eye. As the bows appear in the direction of these deflected rays, 

 it follows that the different colors will not be superposed, and that the breadth 

 of the compound bow Avill be greater than the breadth of the sun by the total 

 amount of their want of conformity. The index for the red may be taken at 

 1.346; that for the violet at l.ooo. Employing these values, we have for the 

 bow by one reflection : 



Violet rays.. £v = 58= 40'. J'^ = 1C9°43'. Radius of bow = 40° 17'. 

 Redrays tr = 59=23^'. J'^ = 137^ 58^-'. Radius of bow = 42° 1§'. 



* The general expression for the deflection beiug — 



A("')=2i-{-/»7r— ::;(?7i-f 1 )p, 

 its differential is (/A(™)^2</t— 2(»«-{-l)rfp ; which, when A('") is a uiiuiuuini, is equal tu zero. 



di 

 From this we obtain the ratio. -~^z^m-\-\. 



From the Sncllian law, sini=Msiup, w being the index of refraction. This furnishes au- 

 another value of the same ratio, since cos«dt;=;Hcospr//>. 



^ di ricos/j 



Or, — -=: =7n-\-\ ; ana (jK-i-l)cosi=^wcos/). 



dp cost 



Squaring this, and adding to it 1 — cos-i^M-sin-'p, mciiiber for member, we obtain — 



[(H(+lj- 1 ] COS'-i + l=::n-i^C0H'^p-f Siu-p)=?i"^. 



From which we deduce the result in the text — 



8 S 





