C. Barns — The Comparison of Two Screws. 337 



For the reasons stated it was necessary to coordinate the 

 successive sections of the measurements made and a fresh 

 fiducial mark was determined in terms of the preceding when- 

 ever the ellipses lost adequate clearness. To find the relation 

 of the two screws, the mean of the initial and final halves of 

 each series was computed by deducting the corresponding 

 observations from each other and averaging the result. In this 

 way the ratio r of the old appears in 



Series I r' = -9962 



" II (mirrors reversed) r" — -9997 



Thus the true relation is finally 



r = (?•' + r") / 2 = -9980 



Usually a large part of the difference of values of the two 

 screws is to be ascribed to the angles of alignment a and a' , 

 if no special means are taken to orientate them accurately. 

 With the ratio r = S / s given, the ratio of the alignment 

 angles would follow, but this is of little value. In fact, if 



, _ - . .2 — a 



a = a + da Cla = (1 — r) 



2a 



where a 2 may be neglected in comparison with 2. Hence da 

 increases as a decreases, numerically. The real problem of 

 finding a for the measuring micrometer is naturally not touched 

 by such a method, in case of two slide micrometers. It is 

 given at once, however, when one of the micrometers is of the 

 form of fig. 2, where a' (say) is necessarily zero. Hence if the 

 screws are identical cos a = 1/r. Virtually, however, S cos 

 a = s is the effective absolute value of the micrometer screw 

 calibrated in this way in terms of s. This is a particular 

 reason for the development of such an apparatus, fig. 2, to 

 serve the purposes of comparison and standardization. 



4. Conclusion. — The advantage of the present method is 

 that steps of any size, quite arbitrarily, are admissible and there 

 is no danger of ever losing count. A rigorously linear slide 

 is presupposed. If the slide is slightly circular, even with 

 very large radius, the ellipses are soon blurred and lost. To 

 restore the ellipses by rotation of mirror is possible, but in any 

 case precarious. 



Brown University, Providence, E. I. 



