long lines

for long lines the effect of curvature (c) and refraction (r)it can be seen that the difference in elevation (∆ H )  between A and B is :

thus it can be seen that the only difference from the basic equation for short lines is the correction for curvature and refraction (c-r)

Although the line of sight is refracted to the target at D the telescope is pointing to H , thereby measuring Although the line of sight is refracted to the target at D , telescope is pointing to H , thereby measuring the angle a from the horizontal .it follows that S sin a = ∆ h = EH  and requires a correction for refraction equal to HD .

The correction for refraction is based on a quantity termed the " coefficient of refraction (K) Considering the atmosphere as comprising layers of air which decrease in density at higher elevations . the line of sight from the instrument will be refracted towards the denser layers .

the line of sight therefore approximates to a circular arc of radius Rs roughly equal to 8R , where R is the radius of the Earth , However ,due to the uncertainty of refraction one cannot accept this relationship and the coefficient of refraction is defined as K = R/Rs

An average value of   K  =  0.15 is frequently quoted but , as stated previously , this is most unreliable and is based on observations taken well above ground level .

Recent investigation has shown that not only can K vary from -2,3 to  +  3,5 with values over ice as hogh as  +  14,9  but it also has a daily cycle Near the ground , K is  affected by the morphology of the ground , by the type of vegetation and by other assorted complex factors .

Although much reseach has been devoted to modelling these effects . in order to arrive at an accurate value for K , the most practical method still appears to be by simultaneous reciprocal observations .

As already shown , curvature (c) can be approximately computed from c  = D 2  /   2R  .  and   as   D   =  S  we can  write ,

c = S2 / 2 R

As the refraction K  =  R/Rs we have 

8/2  =  SK/2R

Without loss of accuracy we can assume JH  = JD  = S  and treating the HD as the are of a circle of radius S :

HD  = S  . 8/2  = S2 K/2R  = r 

(c - r  )  S 2  ( 1  - K  )  2 R

All the above equations express c and r in linear terms , to obtain the angles of curvature and refraction , EJF and   HJD  , reconsider   , imagine JH  is the horizontal line JE  

and JD the level line JF  of radius R . Then 8 is the angle subtended the center of the Earth and the angle of curvature is half this value , To avoid confusion let  ∂ =  ⦵  and as already shown :

⦵/2   =   S/2R   = َ   َ(c  -  rَ )   =  ٍ  ( 1  -  K ) / 2 R rads

Note the difference between equations in linear terms and those in angular .