48 Wadsworth — Simple and Accurate Cathetometer. 



they are roughly proportional to tang <p. If we draw a line, 

 no (fig. 5), inclined at a small angle «, in each direction from o y 

 to the longitudinal line on the scale s s, and read in each 

 case to the intersection of this line with the radial line u 

 on the pointer ; we increase the scale reading by an amount 

 sr = os tang a tang cp, and the new scale reading is therefore 



os ^ c'o tang cp . [1 -f tang a tang cp~\ 



Hence we have only to make tang a tang <p = d, in order to 

 make os the new scale reading directly proportional to J as 

 before. To find the inclination a of the line n, o, to the axis 

 of graduation we have only to put 



tang #tang cp = S 



for some particular value of (p. Suppose we do this for <p = 30°. 



Then we find 



•0071 

 tang a — or a = 42 



Using this value of a to calculate the values of the scale cor- 

 rection, rs, at other points we find the values given in the 5th 

 column of Table I. As will be seen, they differ on the aver- 

 age from the corresponding values of d by less than -^ per 

 cent ; or only about O'OOOS 111111 at the maximum for <p = 25°. 

 By this simple method, therefore, the necessity for making any 

 correction to the scale reading, even in the most accurate work, 

 is entirely avoided. 



The exact constant of the scale reading for any particular 

 value of the index, differing from those given above, may be 

 either calculated from the above formula or determined experi- 

 mentally. 



In order to always make the value of a l mm scale division 



correspond to some convenient fractional part of a mm. the 



support for the scale is made adjustable in height so that the 



fi-—\ t 

 value of c'o may be always made equal to — where a is 



the fractional value desired. One advantage which the rotat- 

 ing plate has over the ordinary eye-piece micrometer, is that its 

 constant remains the same for all distances of the scale from 

 the telescope.* 



The above form of parallel plate micrometer may also be 

 advantageously substituted for the eye-piece micrometer in the 

 first instrument described. In this case it should be placed 



* The great practical advantage of this form of micrometer over the ordinary 

 form is its much greater simplicity and cheapness. For these reasons it would 

 have been adopted in all of the above instruments had it not happened that we 

 already had on hand a number of micrometer eye-pieces which were available for 

 this purpose. 



