VISION THROUGH SMALL APERTURES IN WALLS. 395 
The perpendicular distance, [¢, of J from the mirror M is equal to 
R.cosa+r7, The distance dt is equal to R sin a. 
Tan @=) = et ii Gt A Sin. @ 
: Ti” Cd” Lt+Cd~ Ros a+ 2r 
. = ie AR GN SD. 
ss as fa eae) ~ Reosat+2R cosa+2 
1h n 
Therefore, «= a—tan-!_S"% _. 
cos a + 2 
n 
Ike (0 = 2p 
tan (a—a) =" © tan 4a. (See note below). 
cos a+ 1 a 
Therefore, d C m=a—a#=4 a=a, or a=22. 
Note.—-Illustration of trigonometrical relation between the sine of 
an angle and the tangent of half the angle (figure 6) 
Fia. 6. 
ALC = 1, Sia @== ID IB ; 
——— cosa= CBs tan G)= ee See 
DADS a 2 14+CB 1+40cosa 
Hence, if the distance of the apertures from the axis is twice the 
distance of the mirrors from the axis, the angular displacement of the 
mirrors is twice the angle measured, instead of half of it as in other 
reflecting instruments, and this gives four times the precision of 
observation obtained by the ordinary sextant. 
Other ratios of r and R apparently give varying relations hetween 
a and w as a increases, and this would make it difficult to graduate the 
instrument except for the above proportions. 
At G in figure 1, graduations are shown on the edge of the plate S, 
which is made to project through the side of the box. A mark on the 
side indicates the proper reading, the scale being so placed that it 
reads zero when the mirrors are perpendicular to the line joining the 
apertures. 
Maximum Laur or Anat Measure wirn Revotvina Mirrors. 
The plane of the mirror, in revolving, passes the aperture when 
cosa= (= )= _" and the mirror may reach a distance R from the 
centre C; that is, to the slit. In this case when n=2; cos a=-h, 
a=120; and a=60°, 
