(III.) 



516 Prof. E. Ketteler on the Dispersion of Light 



coordinates z, x of the plane of incidence, they are also written 

 thus : — 



•*%+***%-*($+&)-<>< • • « 



In order to integrate these equations, we put 



A Tlqz f 277-/ / ~, , Zp + X sill A ") ^ 



p= s Ae^cos| T ^-©'+-^— — -)}, 



. .. ?i Q z f 27r/ ~, 2» + #sine\ 1 nA I 

 p' = A'e *' cos« YV "t5 / J ; J 



where, for abbreviation, '&=%' — %; further, 

 a E = e, a D = ^? p = vcosr, sin e = v sin r, p 2 + sin 2 <? = v 2 . 

 After the carrying-out of this substitution, equation I. 

 becomes 



wA 2 cos (/> + 2m'A /2 cos (</>-2A)=m 2 A 2 [(y-2 2 

 + sin 2 e) cos (f> + 2p^ sin </>]. 

 It subdivides into the two following : — 



2 2 1 2m'A /2 cos2A . 2m'A /2 sin2A ^^ 

 " 2 -<Z 2 - 1 = m A 2 — > %*= ^— ^ IV ') 



Now, inasmuch as the right-hand side of these expressions, 

 as characteristic of the medium referred to, is independent of 

 the angle of incidence, we have, in accordance with equation 

 (12):- 



p 2 — g 2 + sin 2 e=v 2 — q 2 = a 2 — b 2 , pq — ab. . . (V.) 



Consequently for the actual variable ratio of refraction v 

 (conf. I c. p. 70), 



2 v 2 = a 2 -b 2 + sin 2 e + \/(a 2 - b 2 - sin 2 e) 2 + 4aW. 



Analogous treatment of the differential equation II. con- 

 ducts to the relations 



(a 2 + 6 2 )VA /2 sin 2A = *'X 2 [(a 2 -6 2 )A /2 sin 2 A 



-2a5A /2 cos2A], 



(a 2 + t 2 ) 2 (eA 2 + e'A' 2 cos 2A) = /c / \ 2 [(a 2 -6 2 )A /2 cos 2 A 



+ 2ab A /2 sin 2A. J 



Combining them with equation IV., and omitting the sum- 

 mation-symbol, we now obtain again directly (that is, without 

 the aid of the complex) the earlier equation (14), and there- 

 with a and b immediately as functions of \. 



(VI.) 



