T4 MEASUREMENT OF ANGLES. 



scale and of the telescope in respect of the mirrors may be verified 

 at each instant by the position of the cross wire on the fixed mirror. 



671. CONCAVE MIRRORS. The methods of observation by 

 mirrors may be modified in various ways. One of the most 

 convenient is to use a concave mirror, placing the scale at the 

 focal distance, and sighting the mirror through a telescope adjusted 

 for parallel rays. 



Let O (Fig. 131) be the centre of curvature of the mirror, F its 

 focus in the initial position, / its focal length OF. The axis of the 

 telescope is in the vertical plane OC, and directed towards the 

 centre C of the mirror. The centre of a scale EE' on the cylinder 

 is at F, and its axis placed vertically passes through the centre C 

 of the mirror. It is clear that if the mirror turns through an angle #, 

 we shall see in the telescope the image of the point X of the scale, 

 such that the arc FX corresponds to the angle 26. All rays from 

 the point X form after reflection a pencil parallel to the optic axis 

 of the telescope, and the image is in the focal plane. The telescope, 

 when once regulated for parallel rays, might then be placed at any 

 given distance ; the only effect of increasing the distance would be to 

 diminish the field. 



In fact, the field being always defined by the condition that at 

 least one of the rays from a point passes through the centre of the 

 object-glass, the visible length / of the scale, adopting the same 

 notation as above, will be defined by the ratio 



/ a f 



/=D' or /= "D> 



we have still / = a, for f D ; but the value of / is inversely as the 

 distance D of the telescope. 



In this case the diameter a of the mirror should be equal to that 

 of the object-glass, if we wish to utilise completely the optical power 

 of the telescope. 



We might, lastly, observe a deflection half that of the limiting 

 angle of the telescope ; for, retaining the same notation, we have 



The conditions are, in fine, the same as if we had a plane mirror 

 with a scale at an infinite distance. 



672. A concave mirror has some practical advantages. In the 

 first place, it is easier to obtain a spherical surface than a plane one, 



