356 PROCEEDINGS OF THE AMERICAN ACADEMY 



If now we put 



d = the number of spaces in the band, 



j'o = the micrometer reading on the first line, 



;'i = „ „ „ „ second line, 



rn= „ " » » n'Mine, 



il^ =z the mean reading of the first d lines, 



3fi =■ „ „ „ „ second d lines, 



JI„= „ „ „ „ (n -\- ly" d Vines, 



nip = the value of 1 rev. at p revolutions, 



we have 



171^ = 31^ + , — 3f, 



when the spaces in the band are commensurate with the value of 

 one revolution. 



We have also the accumulated error, from the p'* to the (/>-{-")'* 

 term, — 



^= (w — ?H^) + (m — m^^i) .... (w — »?^ + „), 



depending on the whole revolutions. 



Tiie value of the corrections to be applied depending on the irregu- 

 larities of single parts of one revolution, will be of the general form : — 



TOp^ ( ^^ ) 



In the present case we have assumed d=lO, and the value of the 

 Bcrew is investigated for each -j'j of a revolution from O.'OO to 25.''0 

 of the scale. 



I am indebted to the courtesy of Prof. ^Y. A. Rogers for a band of 

 ten lines corresponding at 75° F. to one revolution of my screw, ami so 

 equably spaced that the spaces are sensibly uniform with any powers 

 used in the following investigation. I should readily have detected a 

 difference so great as 0.00001 of an inch between any two spaces of 

 the series. 



In making the observations from which the ftUowing results are 

 derived, a i objective by Crouch, having an angular aperture of 100°, 

 and adjusted for glass cover of the slide, was combined with a short- 

 focus negative eye-piece provided with a reticule on cover glass placed 

 in the focus of the eye-lens. The maguifying power was 1050 diame- 

 ters, nearly. 



