BY REFLEXION FROM METALLIC SURFACES. 277 



But observation shows in this case that tan %=45° when |=45° in all instances, 



hence for incidence 0° a=&. 



(18.) More generally, differentiating (15.) in respect of %, we find for any given 



value of ^, that is of a and /3, the value of % for the minimum, or darkest branches, 



^ 2«/3 cos p 



tan2x=- ^2_^/ ^ 



or we might deduce directly 



lan % - - 2«/3 cos g =5= V V "*" 4«^/3^ cos^g^ " 

 Hence we may make the same inferences as before. 



And if when cos f =0 we have also a=j3, or circular vibrations, it is worth while to 



observe that there results 





 tan 2x= Q- 



But in general for the change of plane, or of x, this formula does not assign any 

 precise values, since the form of the function of § involved in a and /3 is unknown. 



(19.) In the absence of any thecyretical law 1 have had recourse to the empirical 

 modification of Fresnel's formula proposed by Professor MacCullagh*'. And avail- 

 ing myself of the computed values which he has given, in the case of steel, for the 

 coefficients of the component vibrations a a' (assuming a'=/3 in my -notation) and the 

 retardations ^ I' (where §— ^'=^ in my notation), I have deduced the values of the 

 product a «', of a^ — a'2, and of cos f, and introducing these in formula (18.), I find 

 the resulting values of x ^^ close agreement with those given by observation. The 

 elements of this computation and its results are exhibited in Table IV. 



Apparatus. 



The general principle of my apparatus will be apparent from the nature of the ex- 

 periments, but it may not be superfluous (especially for those who may wish to pursue 

 similar observations) to annex an outline of the construction, which, after trial of 

 various forms, I have found best to unite the requisite conditions for the purpose in 

 view ; while it is readily convertible into an ordinary polariscope ; besides admitting 

 of the addition or substitution of other parts, when required, for different objects. 



The accompanying Plate II. fig. 1 gives a general view of the arrangement and 

 will suflSciently explain itself: the polarizing part (P) contains a Nicol prism which 

 can be turned in azimuth, measured by a graduated circle ; a condensing lens (L) is 

 also attached to it. 



The analyzing part (A) contains in the eye-piece (E) a lens, a calc-spar, and a tour- 

 maline ; and bears a graduated circle, on which the azimuth of analyzation is read off. 



The metal under examination is placed on the support (R), which can be raised or 



* Reports, Royal Irish Academy, October 1836. 

 2o2 



