Simple arid Accurate Cathetometer. 129 



and carried in a fork which is a prolongation of the boss D 

 shown in the preceding figures. 



The rotation of the plate on its axis shifts the ray from o', 

 which passes through it, and hence also the image of the scale, 



Fig. 4. 



Fiff. 5. 



by an amount A which may easily be shown to be 



r cos <t> n 



A = £sm<£. 1— . . = . 



r L yn 2 — snr<pj 



<£ being the angle of rotation measured from the position in. 

 which the plate is normal to the ray. This expression may be 

 written : 



A=,^tang</>.f(l- , "»» .V^l. (1) 

 n. sy IV ^9i 2 -sin 2 ^/ n-1 J w 



rc-1 



Hang <£./((£). 



The quantity in brackets or /(</>) may be shown to be very 

 nearly unity for all values of <£ between 0° and 30°. In 

 order to determine its exact value, we may develop it into a 

 series as Poynting does, but since this series is only rapidly 

 convergent for low values of <£, it is on the whole better to 

 compute it directly from (1) , which is in a form well adapted 

 to logarithmic computation. I have calculated the values of 

 /(<£>) for values of (j> from 5° to 30° and for two values of n, 

 viz. tt=l*5 and n=l*55, about the mean indices of the glass 



Phil. Mag. S. 5. Vol. 41. No. 249. Feb. 1896. K 



