1877.] 



Magnifying -power of the Half -prism. 



13 



These results might also have been obtained by considering the prism 

 as made up of two half -prisms separated by a thin plate of air. The dis- 

 persion of the first half is — , and this is magnified m times by the second, 



which also adds its own dispersion A,, thus giving 3 . w 1 + A,=2A j . 



This will be seen at once to be equivalent to the algebraic process em- 

 ployed. 



The above results may be summed up as follows : — 



1. The purity of a half -prism magnifying is equal to the dispersion of 

 a half-prism diminishing, and is the product of the dispersive power of 

 the glass and the tangent of the angle of the prism. 



2. The dispersion of a half-prism magnifying is equal to the purity of 

 a half -prism diminishing, and is m times the purity of a half -prism 

 magnifying. 



3. The purity and dispersion of an isosceles prism are equal, and are 

 each twice the dispersion of a half-prism magnifying. 



In future when the letters m, A, and n are used without suffixes, they 

 must be understood as denoting the magnifying-power, dispersion, and 

 purity of a half-prism magnifying. 



Confining our attention for the present to the ordinary form of a 

 spectroscope in which £</>'= — c^' = 0, the most important question to be 

 considered is the irrationality of dispersion, which in fact fixes the limit 

 to the refracting angle of the prism, unless the spectroscope is to be used 

 only for one part of the spectrum. 



The formula for the dispersion is chiefly useful as giving the irra- 

 tionality, which is seen to increase with the dispersion, but in a more 

 rapid ratio. In fact if we put A A and A H for the dispersions correspond- 

 ing to the same small value of — for the Eraunhofer lines A and H. 

 i. e. the values of the scale of the spectrum at those two points, we have 



log ^5 = log tan 4> K ~~log tan ^ A =s ^ ec ^ (^ H — ^ A ) approximately 

 A A tan \p 



sin 2y fx 



In practice, however, it will be more convenient to express the ratio 

 in terms of the magnifying-powers for A and H ; thus 



A A ™> k ' fy A 



if the prism be adjusted to minimum deviation in each case. 



In the following Table the angles of emergence, the dispersioil, and 

 the magnifying-power are given for a series of half-prisms of very dense 



