GRADUATION OF THE SCALE. 



\ve may expand the value of tan 6 in a series. If ;;/ stands for the 



1 x-p 

 expression , we get 



2 a - p 



tan 6 = m [i - m' 2 + 2m*] = ;// - m z + 2 ;;A 



For a deflection of 15 the error made by neglecting the further 



terms of the series is less than . 



10000 



664. If the mirror is a glass one, silvered at the back, or, more 

 generally, if a glass plate is interposed between the telescope and the 

 reflecting surface, so as to be traversed by the rays in a direction near 

 the perpendicular, a correction is necessary for the value of the 

 distance d-p. If e is the thickness, and n the refractive index of 

 the plate, the rays which traverse it behave as if they had passed 



through a layer of air whose thickness is - ; the distance of the 



scale from the reflecting surface should then be diminished by 



e n- i 



e or e . 



n n 



665. GRADUATION OF THE SCALE. The most direct method of 

 obtaining the angular graduation of the scale would be to observe 

 the deflection of the image which corresponds to a rotation of the 

 mirror, measured on a divided circle. If the movable system is not 

 submitted to any other action tending to give it a determinate direc- 

 tion, and if the wire is attached at the top to an arrangement 

 movable over a circle divided horizontally, which has the same axis 

 as the wire, it is sufficient to turn the whole system through a known 

 angle, and to observe at the same time the displacement of the 

 image in the telescope. In the case in which the system to which 

 the mirror is fixed is acted on by an extraneous force, like a magnet 

 in a magnetic field, it will be necessary to fix temporarily the mirror 

 in respect of the rest of the apparatus, or to provide this with a fixed 

 mirror situate at the same distance from the axis as the movable 

 mirror. 



As the tangent of the deflection is given by the ratio of the two 

 lengths, it is sufficient to measure directly in the same unit the scale 

 and its distance. 



The operation, which consists in measuring the distance from a 

 scale to a movable mirror, presents some difficulties when the error 

 has to be less than a ten-thousandth, which corresponds to an 

 approximation of o'i mm. for a distance of one metre. For this 



