BY REFLEXION FROM METALLIC SURFACES. 275 



(12.) This formula is general for all positions of the polarizer and analyzer, and 

 for light of all degrees of ellipticity. If we had taken (as in my former paper) 

 ^^=(45—^), and at the same time supposed a=|3=l (which is equivalent to the as- 

 sumption then made for simplification), the formula (9.) would become at once the 

 same as in that paper. 



The first terms independent of 6 express the intensity of the ground on which, as it 

 were, the rings are formed : those involving ^ vary with the incidence or degree of 

 ellipticity: those involving 6 give the rings; the last only being retained when the 

 ellipticity is a maximum, and disappearing for plane polarization at the incidence 0. 



From these formulas we at once trace all the well-known phenomena of the rings 

 in plane and elliptic light, by following the changes in formula (9.) on advancing 

 into the adjacent quadrants, or supposing -v// to become •4/+90, and by consequence 

 changing also 9 into (p+90; which gives a change of sign in the terms involving 

 sin 2-^, cos 2-4/, sin 2<p, cos 2<p. 



Again, with regard to changes in the analyzer, it is evident that whatever be the 

 value of % if we increase it by 90°, we shall have to substitute in (10.) cos {x—-^) for 

 sin (x—-^) and vice versd ; as well as — sin 2(%— %//) for sin 2(%— %//), and - cos 2(%— 4) 

 for cos 2{x—-^), or, on the whole, the expression is obviously complementary, except 

 the last term. 



(13.) In any case on giving successive values to -vj/ round the rings, at those points 

 where the sines or cosines vanish, the disappearance of any term when accompanied 

 by a change of sign indicates (so far as that term affects the total intensity) a change 

 from dark to bright at that point, or a complementary character in the adjacent por- 

 tions of the image ; when without a change of sign it indicates a simple maximum or 

 minimum of light, in the rings or in the ground, according as the terms affected in- 

 volve & or not ; and for light of different ellipticities according to the value off. 



Thus for the values of % in general, in (11.) — 

 At -J/rirO, or 90°, 



terms 1, 2, 3, for the ground ") .. 



^ „ , . > disappear without change of sign. 



8, for the rings ^ ^^ & & 



At i|/=45, 



4, 6, for the ground *) ,. . , , 



7, 9, for the rings j '^'^''PP'''^ '^'"^ '=''''"«" "^ "S"' 



5, for the ground, disappears without change of sign. 



4, 6, for the ground ■) 



^ r. .t ' r disappear with change or sign. 



7, for the rings j ^^ o o 



In general then there are dislocations of the rings, with more or less complete com- 

 plementary changes in the ground both at the quadrants and half-quadrants. 



At the maximum ellipticity, since terms 5, 6, 7, 8 disappear, there are disloca- 

 tions only at -^=0, with a change in the ground at •4/=45, or a gradual decrease of 

 brightness from one end of the quadrant to the other, which is exactly the appearance 

 observed. In plane polarized light we have the system of eight dislocated sectors, 



MDCCCXLV. 2 o 



