4o L. B. Tuckermam 



kP u -\-Q 1 + cos 2(1,2) [cos 2 (2, 3) — ( 1 — k) cos 2 (2, 3')] 

 -sin 2 ( 1 , 2 ) cos 2ttN cos 2 ^ A N [sin 2(2,3) 



— (I— K)sill2(2,3')] 



K 1 +sin2(i,2)[cos2(2,3)— (1 — K)C0S2(2,3')] 

 -)- cos 2(1,2) cos 2 7r A T cos 2 7r A A T [sin 2 ( 2 , 3 ) 



— (I — K)sill2(2,3')] 



+5 X sin2 7rA A cos2 7rAA r [sin2(2,3) — (1 — K)shi2(2,3')]=o 



Substituting for Q v K x and 5, their values, Qi=P 



1—e* 



i+e' 



cos 2 1 

 1 



1 — e~ 2.6 

 K,=P ^ sin 2^, and S,=P V, the equations become — 



1 ° i-\-e\ ° i-\-e\ 



for a halfshade match : 

 2e, 



i-e\ 

 for a Lippich match : 



-sin2( 1,2 — B x ) tg27TiV=o; 



(70) 



1— e* 



i+*r 



2<?, 



+ COS2(l,2 6j) [COS 2 (2, 3) — (i — k) COS 2 (2,3')] 



— sin 2 (1.2 — ^ 1 )cos2 7rA^cos2 7rA A 7- [sin 2(2, 3) 



— ( 1— K)sin2(2,3')] 



, 1 Vsin2 7rA 7 cos2 7rAA 7 '[sin2(2,3) — (1 — k) sin 2 (2, 3')] =0 



(70 



In these equations there occur only two variables, e x and 

 2^(1,2 — 0J, the rest being constants depending on the rigid half- 

 shade system. There is then a single pair of values of e 1 and 

 2^(1,2 — 0,) for which a complete match exists. Letting this value 

 of ^=tg77, and the corresponding value of 2^(1,2 — 1 )=a, the 

 analyzing system is characterized by -q and a, which are constants of 

 the system and satisfy the equations: 





tg 2 rj -\- sin 2 a tg 2 7r N=o 

 k sec 2T7+cos2a [cos2(2,3) — (1 — k)cos2 (2,3')] 



COS27rAA r , 



(72) 



-sin 2 a- 



COS 2-kN 



[sin 2(2, 3) — (1— k) sin 2 (2,3')] =0 



196 



