70 BULLETIN OF THE UNIVERSITY OP WISCONSIN 



which the axis of the theodolite is deflected from the true 

 vertical. 



Since the level tube turns with the theodolite when the 

 latter is revolved in azimuth, while the positions of the 

 points V and Z remain unchanged, it appears that the 

 angle t must vary directly with the readings of the azimuth 

 circle, since it measures the inclination of the plane of the 

 level tube to a fixed plane passing through the vertical 

 axis of the instrument. If we represent by AO the reading 

 of the circle when the arc VS is made to coincide with VZ, 

 we shall have corresponding to any other reading A' : 



tan p = tan y cos (A A') (1) 



The value of A in any given case may be determined 

 by finding two positions of the instrument, circle readings 

 Ai and A 2 , in which the bubble stands at the same part of 

 the tube. Since the values of p corresponding to these 

 two readings are equal, we must have: 



A -A 1 =A S -A and A = ^(A 1 +A t ) 



If A' and A" denote slightly different readings of the 

 azimuth circle, and &' and b ff the corresponding readings of 

 the middle of the bubble on the level scale, we may write 

 two equations similar to equation (1), and taking their 

 difference obtain: 



sin (p' p") A' A" . / . A'4-A"\. ,~. 



, - 7 = 2 sin - jr - sin I A a -- ^ - I tan y (2) 



cos p cos p 2 \ 2 / 



Since p' p" is the distance moved over by the bubble, 

 we may write p' p"=(b' &")$, where d is the value of a 

 division of the level, and transform (2) into 



2 tany cos*p sin 4 (A' - A") sin L^ ~ * (A ' + 

 sin I- b' - b" 



In this equation cosPp may usually be put equal to 1, or 

 its actual value may be found from the average value of p 

 given by equation (1). Every other factor in the second 

 member of this equation is known with exception of tan y, 

 and the determination of y will determine d. 



