132 A Simple and Accurate Cathetometer. 



to A for all values of <p ; but this, as he himself recognizes, is 

 hardly practicable on account of mechanical difficulties. We 

 may, however, obtain the desired result very much more 

 simply. An inspection of the values of S in Table I. will 

 show that they are roughly proportional to tang <f>. If we 

 draw a line no (fig. 6), inclined at a small angle a, in each 

 direction from o, 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 «r=o« tang a tang <p, and the new scale-reading is 

 therefore 



os^c'o tang <j> . [1 + tang « tang </>]. 



Hence we have only to make tang a tang <f>=$, in order 

 to make os the new scale-reading directly proportional to A 

 as before. To find the inclination a of the line w, o, to the 

 axis of graduation we have only to put 



tang u tang <£ = S 



for some particular value of (p. Suppose we do this for 

 = 30°. Then we find 



tang a= vyy- or a =42'. 



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

 correction, rs, at other points, we find the valnes given in the 

 5th column of Table I. As will be seen, they differ on the 

 average from the corresponding values of S by less than -^ per 

 cent. ; or only about 0*0008 millim. at the maximum for 

 <£ = 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 1 millim. scale 

 division correspond to some convenient fractional part of a 

 millim., the support for the scale is made adjustable in height so 



n — 1 t 



that the value of c'o may be always made equal to , 



J - J - 1 n a 



where a is the fractional value desired. One advantage which 



