482 



Dr. C. V. Burton on a Physical Basis 



is thus decreased by the same amount at each half-swing, 

 until finally a half-swing leaves the index between the limiting 

 positions of friction, where it remains permanently at rest. 

 If the initial displacement (T>) of the index was large com- 

 pared with the range of frictional error ( + <i), we may assign 

 the same probability to all displacements between T> — 2d and 

 D-\-2d ; and since the final displacement differs from D by 

 an exact multiple of 2d, it immediately follows that all final 

 displacements between +d are equally probable, larger errors 

 being impossible. The curve of error will be like fig. 1. 



If friction is greater when the index is at rest, the result is 

 rather curious. Let +d f be the limits of equilibrium under 

 statical friction ; then there will be equal probabilities of 

 errors between the limits 2d—d' and — d r , and also between 

 — 2d-\-d / and +d r . If d' is < 2d, these ranges of error over- 

 lap, and the curve of error is like fig. 3 ; if d! is > 2d the 

 curve is like fig. 4 ; if d! = 2d, we have simply fig. 5. 

 Fig. 3. Fig. 4. 



Fig. 5. 



These results are easily obtained by considering initial dis- 

 placements between the limits 4nd + d / and 4,{n + l)d + d r , 

 where n is an integer ; they refer of course to the actual, not 

 to the observed position of the index. If, while the index 

 was in motion, three successive excursions were read (with 

 perfect accuracy), the inferred position of equilibrium would 

 only be subject to an error due to deviations from the assumed 

 laws of friction. 



5. Now let a declination-needle which is to trace a con- 

 tinuous record be subject to frictional error. If the black 

 line in fig. 6 represent the true declination-curve, the curve 

 traced by the needle will be something like the dotted line. 

 Here the law of error depends on the (variable) friction of the 

 needle, and on the kind of changes which occur in the quantity 



