8 



MEASUREMENT OF ANGLES. 



angles OCN and NCX are equal from the laws of reflection ; OCN 

 is moreover the angle through which the mirror has been turned. 

 If this be called 0, and if x and p are the distances from zero of the 

 divisions in X and O, and if d is the distance CO, we have 



OX x-p 

 (0 tan 20 = - = ^-. 



If the surface of the mirror M, instead of being on the axis 

 of rotation itself, is at a distance /> = CM (Fig. 130), it is easy to 



Fig. 130. 



see that the division X seen through the telescope for a de- 

 flection 0, is the same as if the mirror had been turned through 

 an angle 0, about the point C', and that we have 



(*) 



tan 26 = 



d- 



cos e 



As the deflections are always small, if the distance p of the 

 mirror from the axis is small in comparison with the distance of 

 the scale, we may put unity for cos 6 in this expression. 



For very small deflections we have approximately 



(3) 



2 d- p' 



When the approximation given by this simple formula is insuffi- 

 cient, and we wish to avoid the calculation of trigonometrical lines, 



