44 REPORT — 1872. 



The lengths A C and B D have to one another the same ratio as the velocities of 

 light at A and B respectively ; or 



AC : BD : : v^ : v,. 

 If A B and C D be produced till they meet in G, the length G A is the radius of 

 curvature of the ray at A. Let this radius be denoted by r. Then, since A B is 

 =X sec 6, we have obviously 



i\~v.-, : i\ : : \ sec 6 : r. 



Hence curvature or — = ^Lll'a _ qqq q 



r i\\ 



or curvature cc cos ^ ; wWch shows that the curvature is a maximum when 6=0, 

 that is, when the ray is level, and that the curvature diminishes to zero as the ray 

 becomes vertical. 



The result here brought out, - = tri!? . cos 6, 



is perfectly in agreement with that arrived at in the previous investigation of Prof, 



Purser, namely - = H—— sin /, seeing that sin i is =cos 6, and that, according to 



the undulatory theory of light as confirmed by experimental proofs, it is known 



that v^ : v.^ : : n : I, so that E1III2 n^ugt \^q equal to tZi}. The new method has 



however, the advantage of quite clearing away the perplexity involved in the other 

 by the collapse of the reasoning when brought to the extreme case of the level ray. 

 In the new method no such collapse occurs ; and, in fact, the new method shows 

 clearly how the real fundamental principle (that of retardation of velocity in the 

 denser medium, on which the bending depends, and which holds good quite as much 

 for level rays as for any others) is allowed in the previous in^■estigatiou gradually to 

 fade out of the reasoning, till, in the case of the level ray, it has absolutely vanished 

 from the conditions which were taken into account. The previous method, like the 

 modes of considering the subject of atmospheric bending of rays which appear to 

 have been most generally entertained hitherto, took a consequence of the important 

 fundamental principle into account instead of the principle itself (that consequence 

 being the proportionality of sines of angles of incidence and refraction in case of 

 oblique transition of ligli't from one lamina to anotlier of different density) ; but that 

 consequence happens to be not so general as the principle from which it'foUows, and 

 to be one which becomes nugatory or non-existent in the case of the level ray. 



In concluding, the author wishes to state that it seemed to him rather unlikely 

 that so simple a view of the influence of the atmosphere in effecting the bending o"f 

 rays of light as that which he has now offered could be quite new. He thought 

 that others better acquainted with the science of light than he is must most probably 

 have entertained the same or similar views. He has therefore made inquiries as to 

 the views which have hitherto been put forward regarding the bending of light in 

 the atmosphere and in other mediums of continuously variable index of refraction, 

 or, as they may be better considered in the present investigation, mediums 

 of continuously varying light-velocity*. Much has been written on the sub- 

 ject in general, and on various particular cases of its application ; and views 

 very similar in principle with those here offered appear in various ways to 

 have been entertained, or implied more or less explicitly ; but he ha-s not learned 

 of anything having been taught which has anticipated the treatment of the subject 

 at present offered so as to deprive it altogether of novelty and interest. The subject, 

 he believes, has been very generally considered under imperfect views ; and he will 

 think a good result will have ensued if his drawing- the attention of the British 

 Association to it will serve to elicit from others notice of the best views that ha\o. 

 hitherto either been fully published, or have been entertained or discussed without 

 complete publication. 



Postscript.— From Professor Clerk Maxwell I have learned that, in December 

 1851 or 1852, when on a visit to my brother, Sir William Thomson, he had in his 



* — might be called the index of light-velocity. 



