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XI. On the Conditions which Determine the Ultimate Optical 

 Efficiency of Methods for Observing Small Rotations, and on 

 a Simple Method of Doubling the Accuracy of the Mirror 

 and Scale Method. By F. L. 0. Wadsworth*. 



IN a paper on the " Optical Comparison of Methods for 

 Observing Small Rotations," Rayleigh discussed some 

 years ago the relative limiting accuracy of the two methods 

 then commonly in use for observing small angular deflexions 

 of a suspended system ; i. e., the Gaussian method of scale and 

 mirror, and the method of the pointer and microscope. The 

 conclusion was reached that theoretically the two systems 

 were on a par with each other when the length of the pointer 

 was equal to the diameter of the mirror |. 



It would therefore seem that the latter method has a con- 

 siderable advantage over the former, since for a given weight, 

 or, what is more important, for a given moment of inertia, it 

 is possible to make a pointer of a length of at least ten times 

 the diameter of a mirror, and thereby attain with a given 

 time of swing a tenfold greater sensitiveness. But notwith- 

 standing this apparent theoretical superiority of the microscope 

 and pointer, the mirror is in practice nearly always preferred, 

 not only because of its greater convenience, but also because 

 experiment has shown that it is really not so inferior in 

 accuracy as the above comparison would lead one to expect. 

 The reason for this is that the conditions assumed by Lord 

 Rayleigh as a basis for this comparison do not always hold. 

 Let us consider first what these conditions are. In the case 

 of a revolving mirror Rayleigh assumed that the smallest 

 angular motion which would be perceptible through its effect 

 on a reflected image would be (supposing the mirror rect- 

 angular and the incidence nearly normal) such that " one 

 edge of the mirror advances %\ (while the other edge retreats 

 to a like amount)/'' and thus "introduces a phase discrepancy 

 of a whole period where before the rotation there was complete 

 agreement." This gives us for the limiting angular motion 

 or the expression 



a '=2&' C 1 ) 



where b is the length of the mirror perpendicular to the axis 



* Communicated by the Author. 



t " Optical Comparison of Methods for Observing Small Rotations," 

 Lord Rayleigh, Phil. Mag., Oct. 1885. See also art, " Wave Theory," 

 Eric. Brit. vol. xxiv. § 13. 



G2 



