of Methods for Observing Small Rotations. 89 



it is important not to make the distance either between the 

 double cross-wires or the lines on the scale too small, nor on 

 the other hand too large. When we know the law of varia- 

 tion of A with 8 we can determine exactly what separation 

 would give the best result in a given case. Let us suppose 

 as before that the conditions are such that the final (retinal) 

 images of object and reference-wire have the same width. 

 Then when 8 is zero the two outer images merge into one 

 and we have the case previously considered, in which we 

 found that the smallest observable movement under the 

 most favourable circumstances was that which produced a 

 separation 



2 = 0-9«. 



Suppose next that the two outer objects are separated by 

 an amount equal to about twice their own width, as in fig. 2. 



Fig. 2. 



«- — — — g=4«. X 



The central image, if midway between the outer two, would 

 be clearly separated from each of them, and it would be easy 

 to determine its position to within, say, \ the whole interval 

 8, i. e., to within about 0'8 a, a somewhat higher degree of 

 accuracy than before. When the separation 8 is made 

 greater (say five times the width (2a) of the images, as in 

 fig. 3), experience shows that it is possible to locate the 

 central object to within -J Q to Jq the interval, /. e., to within 

 ^ to \ the resolving power of the instrument*. With still 

 greater separation the accuracy may be slightly, but only 

 slightly, increased. It is obvious that the narrower the 



* This is on the assumption that the position can be located by 

 micrometric methods, one object being made to bisect the interval be- 

 tween the other two. If simple eye estimates only can be made, the 

 accuracy will be less, probably at most not more than ¥ V the separation 

 or \ the width of the image. In such a case better results would 

 generally be secured by making S smaller, 



