CHROMATICS OF POLARIZED LIGHT FORMULAE. 



207 



^'P-^^ 



feronces may be slight, but slight values many times repeated become largo 

 values at last; so that two red rays whose phases are for several undulations 

 sufficiently unlike to conflict, may, after a larger number, be nearly enough alike 

 to conspire. If the numbers 21 and 22 represent the lengths of two undulations 

 of green, after a retardation of eleven times the length of the former, the latter 

 will have fallen half an undulation behind it. Thus, after a certain amount 

 of retardation is reached, there will be found rmdulations of all colors im- 

 part irdl}' distributed through all varieties of phase, and the chromatic phenomena 

 above described will cease. 



A general expression for all these phenomena 

 may be found as follows : Let PP' be the plane 

 of jjolarization of the original ray ; QQ' the princi- 

 pal plane of the lamina; 1111' the conjugate plane; 

 1)0' the principal plane of the analyzer, which we 

 will suppose to be a doubly refracting prism or 

 rhomb of Iceland spar; and EE' its conjugate 

 plane. Draw PA, PB, perpendicular to QQ' and 

 IIR'; BE, BG perpendicular to EE',00'; and AD, 

 All perpendicular to EE', 00'. Then if CP rep- 

 resent the velocity of molecular movement in the 

 original ray, C A and CB will represent its equiva- 

 lent components in the directions IMl and Q'Q. 

 If these components be further decomposed in 

 the directions 00' and EE', we shall have the original velocity CP represented 

 by the four elements CCr, GII in the principal plane, and CD, CE in the con- 

 jugate plane of the analyzer. 



Ilepresent the original velocity CP by V. Put the angle PCQ=«, and the 

 angle PC( )— j. Then the angle OCQ w'iU be a— ,3. The triangles PC A, PGB 

 giveCA=Vsina; CB=Vcosa. And the triangles ACD = ACH, and BCE=BCG, 

 give CD=Vsinacos(« — p') ; CF = Vcorjasin(a — ,3); CG=Vcosmcos(« — 8) ; 

 CH=Vsin«sin(a — ^s). Then, to find the resultant of CG-, ClI, the molecular 

 velocities of the two rays emergent in the plane 00' — that is, of the emergent 

 ordinary ray — we recur to the general equation — 



A^ = ci^ + rt'2 + 2aa'cos0. 



in which we must substitute for a and a' the values of CGnnd CII given above; 

 and for t^he amount of retardation in phase of one of the rays behind the 

 other in passing the lamina, which, if /i represent the actual difi'erence in length 



of path, may be represented by 2--. The equation just stated may be con- 



A 



veniently transformed by adding 2aa' — 2aa' to the second member, wlum it will 

 become — 



A2 = a^^a'^^2aa'—2aa' + 2aa'co8f}. 



Or A^ = {a + a'f—2aa'{l—cosO) = (a-\-a')^—iaa'fimVjO. 



Substituting now the values of a, a', and 0, we obtain — 



A^rziV^I [cosacos(a — /5)-|-sinc(sin(a — fi)\- — 4sinacosasin(r/ — /5)cos(ct — /5) 



^0' 



Which may be reduced to the following entirely equivalent forms : 

 A2 = V2 TcosV + fcos2(2«-;^i) -cos^.Sjsin^/ 



A2=: V^ Tcos^/J - [^m\2a-l3) - sin^/Sj sin^/^ 1 . 



[37.] 

 [38.1 



