370 PROFESSOR CHALLIS, ON A THEORY OF LUMINOUS RAYS, ETC. 



(t) ds ^ , .1 . J "'^ 27r ^ . ^ ,^ dfh , . 'iTT , , 



So« = 2l. — . But we have seen that (^ = cos-— (af-z). Hence -^=-wasin — (o#-^) 



S dy Stt \ rf/ \ 



7HO.' . 2 7r , (h \a'- Stt 



= - o'.V. Therefore .V = — r sin — - (o < - x), and i = ^ . cot — (at - x). Call this quan- 



a- A A 27r« \ 



rfs di 



titv <!>• Then ?< = <I) -— , and u = t^ — - . Let now s = cr, + ctv. Then 

 d,v ay 



and as « in equation (l6) is arbitrary, we may have separately, 



d-CTi dVi , ., 

 d,v^ dy' 



d-(T2 dr<ji „ 



and -—^ + — — + kn-c, = 0, 

 d.f- dy' 



and consider these equations to apply to two distinct rays. At the same time, since s = tn + u^, 



ds ^ d(Ti ^ da-2 , ^ ds ^ do-, ^ das , ■ , .. i , . . . , 



tl) — ^^ = <p h <p — , and <P — = (p H <P ; that is, the sums or the velocities in the two 



d.r dx div dy dy dy 



rays resolved in the directions of the axes of co-ordinates are equal to the resolved parts of the 

 velocity of the original ray in the same direction. Similar reasoning would have applied if we had 

 assumed « = o-i + tTa + cr, + &c. The general conclusion we may now draw is, that a ray may be 

 conceived to be composed of two or more rays in the same phase of vibration, and that if, after a 

 ray has been separated into distinct rays, the parts be put together in the same phase of vibration, 

 they will compose the original ray. 



The foregoing Theory of Luminous Kays, conducts to a very simple and satisfactory explana- 

 tion of the pha;nomena of Polarized Light, which I propose to bring before the notice of the 

 Society at a future opportunity. 



Cambridge Ohservaluri/, 

 Mai/ 11, 18+6. 



