52 L. B. Tuckerman 



Differentiating with respect to e o , and introducing the condition 

 for a perfect match, e=o: 



de„ 



A/ 1 2sin2(i,2)sin27rAA T 



I m J i -f- cos 2(0, 2 — 6 o ) -\- 2 sin 2 (o, 1 — g ) sin 2 ( 1 , 2 j sin 2 tt AiV 



Then the maximum possible error in e o , compatible with an ob- 

 served match, is : 



1 1 -f- cos 2(0, 2- — $j 



| + 2sin2(o, 2 — 1,2 — o )sin2(i,2)sin 2 irA.N 



K=%fV.>*.P>y> ■ • •)■ 



:#/(7>,£y., . . . ) 



sin 2(1, 2 ) sin2 7rA.V 



1 -j-[i — 2sin 2 2( i,2)sin 2 7rAA r ]cos2(o,2 — o ) 

 H-siii4(i,2) sin 2 7rAA r sin 2 (0,2 — 6 o ) 



sin 2 ( 1, 2) sin 2 7r AA r 



(96) 



The numerator of this expression is ~- Although the form of 



f(f m ,a,(3,y ) is not definitely known, it is known that 



for all useful values of Z m , the expression / f (7 m ,a,/3,y, . . . .) 

 decreases with decreasing Z m . If, therefore, the numerator of the 

 expression for 8e o , be decreased without altering the denominator, 

 Se o is decreased and the sensibility increased. This can be done 

 by varying 2^ (o, 2 — Oj. Differentiating the numerator with re- 

 spect to ^(0,2 — 6 J and equating to zero. 



— [1 — 2 sin 2 2 ( 1,2) sin 2 7rAA~] sin 2 (o, 2 — u ) 



+sin4(i,2)sin 2 7rAA r cos2(o, 2 — Q )=o 



Solving for tg2(o, 2— u ): 



sin 4 ( 1,2) sin 2 7rAA r 



tg2(o, 2 — V )- — 



S V "' I— 2Sin 2 2(l,2)Sin 2 7rAA 



from which (97) 



sin 4 (A, 2 )sin 2 7rA A^ 



sin 2 (0,2 — 6 o )- 



] 1 — sin 2 2( i,2)siir2 7rAA' 

 208 



