88 Prof. F. L. 0. Wads worth on the Optical Efficiency 



where o- is angular width of the lines and a, as just defined, 

 the resolving power of the aperture through which they are 

 viewed. The value of X is slightly less for very small values 

 of a than for <r=0. Thus the minimum value of X occurs 

 when 



2( V2 + 1) 5 v 



For this value of a 



It is therefore an advantage that the cross-wire and reference- 

 lines have a certain small finite width, not only on account of 

 greater distinctness but also because, as long as the width 

 does not exceed the value given in (4), they are more easily 

 resolved, i. e., a smaller angular movement of one with 

 respect to the other can be measured. 



II. Suppose as before that both objects are of the same 

 character but are not superposed. The smallest relative 

 movement that can be detected in this case depends upon a 

 number of things, amongst others the initial distance between 

 the centre of the image and of the wire or reference-line ; 

 the width of the image and wire ; and to a greater extent 

 than in the preceding case on the ability of the observer. In 

 any case of this kind it adds greatly to the accuracy of the 

 determination to have either two reference- wires, one on each 

 side of the image, or two images moving together on each 

 side of the single fixed reference-wire. The first condition 

 corresponds to the case of locating the position of one engraved 

 line on a bar (or narrow pointer) in a microscope by means 

 of the double (parallel) cross-wires ; the second to the case of 

 locating the lines of a scale seen by reflexion from the 

 revolving mirror with reference to a single fixed cross-wire 

 in the eyepiece of the observing telescope. In either case, if 

 the central object is nearly midway between the outer two, 

 and if these latter are sufficiently far apart in comparison 

 with the width of the former, it will be possible to estimate a 

 shift of one system with respect to the other of from ^ to 

 gl^ the whole interval 8 between the two outer objects. 

 The ultimate accuracy therefore depends directly on the pro- 

 duct of two factors, the absolute interval 8, and the percentage 

 error of subdividing this interval, which we will call A. As 

 8 increases A diminishes, at first slowly, then more rapidly, 

 and then more slowly again. The product of 8 and A will 

 therefore be larger for either a very small or a very large 

 value of 8 than it will be for an intermediate value. That is, 



