Lord Rayleigh's Investigations in Optics, 271 



Let A B (fig. 5) be a plane wave-surface of the light before 

 it falls upon the prisms, A B the corresponding wave-surface 



Kg. 5. 



for a particular part of the spectrum after the light has passed 

 the prism or after it has passed the eyepiece of the observing- 

 telescope. The path of a ray from the wave -surface A B to 

 A or B is determined by the condition that the optical distance, 

 represented by J fi ds, is a minimum ; and as A B is by suppo- 

 sition a wave-surface, this optical distance is the same for both 

 points. Thus 



JVds(forA)=j^s(forB) (2) 



We have now to consider the behaviour of light belonging 

 to a neighbouring part of the spectrum. The path of a ray 

 from the wave-surface A B to A is changed ; but in virtue 

 of the minimum property the change may be neglected in cal- 

 culating the optical distance, as it influences the result by 

 quantities of the second order only in the change of refrangi- 

 bility. Accordingly the optical distance from A B to A is 

 represented by J (ft + hfi)ds, the integration being along the 

 path A . . . A; and, similarly, the optical distance between A B 

 and B is represented by \ (/i- + Bfi)ds 9 where the integration is 

 along the path B . . . B. In virtue of (2) the difference of the 

 optical distances is 



ffytds (along B . . . B) — §8fj,ds (along A . . . A). . (3) 



The new wave-surface is formed in such a position that the 

 optical ; distance is constant, and therefore the dispersion, or 

 the angle through which the wave-surface is turned by the 

 change in refrangibility, is found simply by dividing (3) by 

 the distance A B. If, as in common flint-glass spectroscopes, 

 there is only one dispersing substance, \ 8/u, ds = 8{A . s, where s 

 is simply the thickness traversed by the ray. If we call the 

 width of the emergent beam a, the dispersion is represented 



by BfM — — —, s ± and s 2 being the thicknesses traversed by the 

 a 



extreme rays. In a properly constructed instrument s x is 



