326 



♦ KNOWLEDGE 



(it all. The ray 6 c' having suffered less deflection than 

 the ray a c, these rays, which had already been slightly 

 divergent when they fell on the air at A and B, are now 

 become more divergent, and proceed from a virtual focus 

 situated as at /, only much farther away relatively from 

 the earth. Similarly the rays S' A', s' B' after refraction 

 along the curved courses A' a' and B' &', emerge on the 

 tracks a! c and i'c', as from a focus (virtual) at /'. 

 Taking all the rays from the given point on the sun, 

 iuclined in the same degree as S A, sB (or rays between 

 these), and S' A', s B' (or rays between these) to the line 

 from that point to C, c, and c', we find a ring of foci, 

 such as f and /'; (to avoid confusion, the ring is 

 not shown as the ring Zr h' is). In other words 

 the point is transformed into a ring. But of course 

 it would only be seen as a ring to an eye suitably 

 placed on the axial line C c c' .) The case corresponds to 

 I hat of the sun's centre supposed to be centrally- behind 

 the earth, in my former treatment of the problem.) Any- 

 where else within the region into which the rays are 

 deflected, the point would be seen either as one point, or 

 as two points on opposite sides of the sun. Thus to an 

 tye at p, one point would be seen as at / close to the 

 earth's edge, and no other ; to a point at q one point 

 would be seen as at /, slightly above the earth's surface, 

 ,ind no other ; to an eye at p', where the rays a c, V c' 

 cross, one point would be seen as at / close to the earth's 

 edge, another as at/' slightly above the earth's surface, 

 but no more. To an eye at c the point would be trans- 

 formed into a focal ring close to the outline of the earth's 

 disc ; while to an eye at c' the point would also be trans- 

 formed into a focal ring, but it would not be quite close 

 to the earth's outline. 



This we had already recognised (see my former papers) 

 in another way. 



The above explanation shows also in what degree the 

 amount of light received from the sun will be diminished. 

 For in whatever degree the already slightly divergent 

 beam of light s B S A is made more divergent at its emer- 

 gence in the form of the beam 6 c' ac, its illuminating 

 power for any surface exposed to it is correspondingly 

 reduced. But since it maj' be shown that the apparent 

 area of any small portion of the sun's surface is diminished 

 when thus seen through ^-^r air, in precisely the same 

 degree that the area-divergence* of the beams of light from 

 it is increased, it follows that in this case as in all such 

 eases, the intrinsic lustre of the surface is not affected 

 except by absorption. 



I incline to think that the fundamental error in Mr. 

 Williams's inquiry into this matter has lain in the sup- 

 (josition that rays undergoing horizontal refraction at the 

 surface of a sphere, that is falling on it tangentially or 

 grazingly, pass in almost as grazingly as they enter, and 

 then suffer total reflection. There is a passage in 

 Brewster's " Optics " which I remember perplexed me 

 very much when I read that work as a boy, in which he 

 speaks of rays under certain conditions being totally 

 reflected inside a sphere and afterwards undergoing 

 repeated reflections and never getting out. I forget 

 the actual wording ; but I am inclined to think there 

 was a real mistake on Brewster's part, not misun- 

 derstanding on mine. (The passage was only a casual 

 remark.) Anyhow it is certain that no light which 

 can get into a sphere of any substance can possibly 

 suffer total reflection inside the sphere — for the simple 

 reason that it can only get in at an angle not exceeding 

 the critical angle, even if it falls tangentially; and at 



• As distinguislicd from 



ir di\ergence. 



angle it enters, at the f 



angle must it 



Fig.-,. 



Thus if RA, Fig. 5, is a ray incident tangen- 

 tially on the sphere A B K at A, refracted according 

 to the law of sines* in direction AB, it will emerge 

 tangentially at B in direction B R'. For, producing the 

 tangents R A and R' B to meet at T, it is obvious that 

 the angle B A T is equal to the angle A BT'; and as 

 B A T is the critical angle so also is A B T. The ray 

 then which has got in, will get out again, along t.ingent 

 B R'. Some light will of course be lost by reflection at 

 B, just as some was lost at A. There always is reflec- 

 tion where there is refraction. Bitt total reflection only 

 occurs beyond the critical angle. 



Thus we can always get light through the very edge 

 of a sphere even of flint-glass or diamond. (Any one who 

 doubts can get a sphere of diamond and try, forwarding 

 to me afterwards as a jircseut for pointing this out — a 

 one-inch sphere will do very well). The experiment can 

 be tried with a globular decanter full of water. (T/uV 

 need not be forwarded to me ; but indeed that wou.ld 

 follow at once from reflection. With a large globe, as a 

 globular fish-bowl, the eye can even be so set that a 

 distant luminous object — as the sun — would be altered 

 into a ring by tangential refractions (an opaque bodj- 

 filling up the middle of the globe will prevent the direct 

 image of the sun being seen at the same time, as it 

 otherwise would ; only the whole ring could not be seen 

 at a single view, forming too large an optical field). 



Figs. 6, 7, 8, 9 show the course of tangentially incident 

 parallel rays : for water (refractive iudex 1-3.3), Fig. 6; 

 glass with refractive index 1-5, Fig. 7 ; glass with 

 refractive index 2 (very heav^- flint-glass) Fig. 8; and 

 air, supposed of uniform density, in i-aciio, with refractive 

 index (O A : L) = 1-00028 Fig. 9, only A B ought to 

 be much shorter, being' really an arc of only 3"^ 10'. 



An eye placed anywhere along B T in any of these 

 cases will get light tangentially from B, in fact sec the 

 remote source of light (supposed to be far away on 

 the left) in the direction T B. An eye anywhere along 



* There is something to my mind rather clumsy in Brewster's 

 way of dealing with refraction tlirougli a sphere. Tiie little circles 

 at incidence and emergence — as I remember them- introduce quite 

 unnecessary complexity. Thus in the above case all we have to do 

 is to describe a circle L D F with radius L such, that ratio A : 

 L = ^ : 1 (/J being the refractive index) and to draw a chord 

 ALB tangent to this circle : A D is the direction in which the ray 

 R A is refracted. Sec. 



