215 



[Hagen. 



the west end of the rotation axis above the true horizon, 90° — k the azi- 

 muth of this west end and 90° + c its angle with the line of collimation, 

 Bessel's formula is 



T =: m -f- n tan S -\- c sec d, 

 and finally, Hansen's formula 



7 = b sec e -|- n (tan d — tan <r) -|- c sec ,J, 

 where n denotes the declination of the west end of the rotation axis and 

 90° — m its hour-angle. All these constants are in the following relations 

 to each other : 



n = b sin ^ — k cos ^^ b= n sin ^ -f m cos ^ ") 



m = b cos c; -f k sin ^ k = — n cos ^ + m sin ^ ) ^ ■' 



For observations by reflection the constant b and consequently m and r> 

 are to be changed, say into h\ m\ n', by the following formulas : 

 V = — 3 /S — b =b — d 



m^ = m — 3 (/9 + b) cos ^ = m — d cos ^ 

 n^ = n — 3 (/? + b) sin ^ = n — d sin ^. 



Hence the three formulas of Mayer, Bessel and Hansen become for obser- 

 vations by refl,ection, 



cos ((T — d) sin (w — d) c 



_l_ k — ' -4- 



' cos d ~ cos o 



cos (<f — o) 

 r=m-\- n tan (J + c sec (J — d ^^ 



T = (b — d) sec ^ + (n — d sin ^) (tan <5 — tan ^) -f c sec ,5. 



As to determining the constants of these formulas, it will be seen, as in 

 case of the azimuth instruments, that they cannot be found, unless the 

 time of the place be known. First we will find the constant d, which may 

 be done in two difierent ways, viz : by the striding level, which, being 

 itself inclined to the true horizon by the angle /?, cannot give the value of 

 b, but it gives the value of 



or by observing the transits of the direct and reflected image of a star. 

 Let T and T^ be the mean values of time for all the transits reduced to the 

 middle thread for direct and reflected image, J T the clock correction on 

 sidereal time and a the star's apparent right ascension, then is evidently 

 a = T-fjT-j-T, hence 



cos(y5 — d) sin (^ — 8) c 

 for direct image a =T + J T + b cos ,J + ^ ~^co^^~" + ^^T^ 



cos (w — 3) sin (o) — d) c 

 "reflect " a = T^ + J T+ (b-d) -^^^^-^ +k ^^^ -f ^^ 



and by subtraction 



y = ,j + b = 3 "- ^- (^-T^) (13) 



