12 MEASUREMENT OF ANGLES. 



rectangular aperture equal to the diameter of the object-glass, its 

 length at right angles to the direction of the lines, and as narrow 

 as is compatible with the illumination of the scale. The images 

 of the figures are not so good, but the lines seen far more pure. 

 668. If the distance of the lens does not increase the sensi- 

 tiveness, it modifies the number of divisions of the scale which 

 can be seen at the same time. We may define the practical limit 

 of the field by the condition that the rays from a point shall cover 

 at least half the object-glass. According to this, if / is the length 

 of that portion of the scale of which the image is seen, we shall 

 have 



da 



With the ordinary arrangement in which */=D, we have l=2a; 

 the visible length of the scale is then equal to double the breadth 

 of the mirror; that is, is equal to the diameter of the object- 

 glass, if the mirror is the minimum size. 



We have implicitly assumed that the angular value - - of 



a + D 



the field thus defined is smaller than the optical field of the lens, 

 as is ordinarily the case ; if the mirror were very large, the size of 

 the field would only depend on the telescope. 



669. The method of the mirror does not enable us to measure 

 large deflections ; it is not convenient to estimate more than 5 

 on each side of the position of rest, or a total angle greater than 

 10. As the image has twice the displacement, the apparent angle 

 of the scale as seen from the mirror should be at least 20; the 

 total length must be 0*3 of its distance, which makes 40 centimetres 

 for a distance of i metre. 



If, when the mean value of the deflections is 3, we wish to 

 estimate the ten-thousandth, the accuracy of the reading should be 



3.60.60" 



or about i"; in order that the angle of penetration shall 

 10000 



be below this limit, we must have an object-glass 16 cm. in 

 diameter, and a mirror 8 cm. As to the reading of the divisions, 



it must be made to within - at the distance ^+D; that is 



2OOOOO 



to say, to o'i mm. if d= D = 10 metres. 



With larger scales it would, be difficult, without greatly increasing 

 the distance, to observe simultaneously the centre and the extre- 

 mities, without modifying the focussing of the telescope. In that 



