41 . 



In the above figure, 



TH = the transit horizontal sight line. 

 The angle HTB = the angle of elevation of the telescope to the foot of the rod 



= E. 

 " " BTA = the angle subtended by any number of revolutions of the 



graaienter screw = G. 

 AB = the length of the rod included by the angle G, when the rod 



is vertical = R. 

 CB is drawn perpendicular to TB. 



Then, CBA = BTH = E TAH = 90 (E-j-G) 



BC = (90 (E + G)J cos E cos G sin E sin G. 

 AB sin (90Vf G) cos G. 



.-.BC = R (cosE tanGsinEO 



tan G=-^- where h is the height above a horizontal line, subtended by 

 one revolution of the gradienter screw at a distance a. 

 n is the number of revolutions made in any given case. 



BT = ^r BC = R^r (cos E sin E) 

 nh nh ^ a 



>.-.BT = R QLcosE 

 and 



and 



HT=BTcosE 



.-.HT=R -cos'E %sin2E . ..,. . H. 





Formulas I and II are general formulas for any gradienter screw. In C. L. 

 Berger & Sons' transits the screw is cut and placed so that when a= 100, for 

 n =2 and h = $, by substitution these formulas become, 



BT = R (100 cos E sin E.) 



HT = R (100 cos 2 E y<z sin2E.) 



Where BT = the direct distance from the center of the horizontal axis of the 



transit to the foot of the vertical rod. 



HT = the horizontal distance from the center of the horizontal axis of the 



transit to the plumb line dropped from the foot of the vertical rod. 



R = the space included on the vertical rod by two revolutions of the 



gradienter screw. 



E = the elevation of the foot of the rod above the horizontal sight line 

 of the telescope. 



When the angle becomes an angle of depression instead of elevation, then the 

 point B is the upper end of the part of the rod used, A B. The distance B T in 

 this case is the direct distance between the center of the horizontal axis of the tele- 

 scope and the upper reading of the vertical rod in the valley. 



The distance HT is, as before, the horizontal distance between the center of 

 $he horizontal axis of the telescope, and the plumb line prolonged in this case 

 upwards from the upper end of the vertical rod. The plumb line in all cases coin- 

 cides with the direction of the rod. 



By means of the following table, it is only necessary to multiply the factor 

 apposite the angle of elevation, by the space included upon a vertical rod by two 

 gradienter screw revolutions, to obtain either the direct or horizontal distance of 

 the center of the horizontal axis of the telescope from the foot of the rod ; or the 

 tame distance from the upper reading of the vertical rod in the case of an angle of 

 iepreseion. 



