184 PROCEEDINGS OP THE AMERICAN ACADEMY 



a lubricant. Without doubt, a part of the discordance is due to errors 

 of measurement, but recent experience has convinced me that, when 

 oil is used as a lubricant, every precision-screw has a variable periodic 

 error, depending on the position of the nut in the line of its motion, 

 especially when there is a decided change of temperature. It is suffi- 

 cient to say here, that, since adopting a substitute for oil, the errors of 

 the screw have remained practically constant. 



(/) It will be seen from simple inspection that nearly all of the 

 errors under discussion are periodic in their character. This may be 

 shown conclusively in the following way : — 



If we have a series of errors depending on one revolution of the 

 screw, which are strictly periodic in their character, and from which 

 all accidental errors are excluded, the given series can be represented 

 exactly by a series of equations of the form, — 



n = -\- a sin x -J- b cos x -\- a' sin 2 x -J- V cos 2 x, &c. ; 



in which 



n represents any given value of the series ; 



x is an aliquot part of one revolution of the screw ; 



a, b, a', b\ are unknown coefficients. 



If, therefore, an expression of this form is found which will repre- 

 sent all the given errors of a series, it may be safely affirmed that the 

 errors themselves are entirely periodic in their character. 



Making n equal, successively, to the mean of the values of the in- 

 dividual errors of spaces 1, 2, 3, &c. of the first three bands of Plate I., 

 we have a series of equations whose solution by least squares will give 

 the normal equation, 



n = — 26.6 sin x -\- 5.9 cos x — 0.1 sin 2 x -)- 0.8 cos 2 x. 



In like manner we get from the mean of the separate values of 

 Bands IV., V., VI., VII., VIII. of the same plate, 



n = — 29.1 sin x -\- 9.9 cos x — 2.0 sin 2 x -J- 1.3 cos 2 x. 



Substituting in these equations the known values of x, we get 

 values of n which are directly comparable with the observed values 

 given in the tables. The observed and the couqiuted values are given 

 in the following table, as well as the deviations of both the individual 

 and the accumulated errors from the computed values. 



