Prof. Encke on Transits. 277 



to be used at the same time, one may choose between the com- 

 binations of m or n with i or k. These combinations, after eli- 

 minating from the equation (A) the proper quantities by means 

 of the relations above given, give the following expressions : 



2 sin ^T . cos {\ T—m) cos n 

 = smi. sec 4) — sm n ( ^ ^^^^^ ) + sm c . sec S 



^ \cos i, cosip / 



/ COS (ip— S) \ . s, 



= sxn^.cosz.cosec(p+ sm w (^^^^^y^j— ^+ smc.secb 

 = sinf.'^^ + sin?«.cos?J ( ^'" ^'''T \+ sinc.secS 



sin(p \cos S . sm (p / 



= — sin k . cos i — ^ — f- sm m.cosnf — 5^ — - ) +sm c . sec S. 



cos ^ \COS S . COS ip / 



The first of these formulee, the one which M. Hansen has 

 chosen, is the most convenient. It may be thus represented: 



T =z i sec (p — sin « . sin 2; . sec <p . sec S + c . sec 8 



It contains only the quantities i, n, and c, which may be di- 

 rectly determined by observations. It shows also most clearly, 

 that with equally good instruments, and consequently equal 

 uncertainty in the values of i and 7i, the uncertainty of the de- 

 termination of time increases with the altitude of the pole. 



If the observation is made at one of the lateral wires, the 

 distance of which from the point for which c has been deter- 

 mined, is called =f, taken positively in the same sense as c 

 above, and if we call t the hour-angle which is to be added to 

 the observation at the lateral wire, in order to reduce it to the 

 meridian wire, we have the two equations : 



sin (c +/) = — sin 8 sin n + cos 8 cos n . sin (t+T—m) 

 sine = — sin 8 sin n + cos 8 cos n sin {r—m) 



• /. » C cos (C + J /■) COS J < ■> 



whence sm t = sm/sec 8 \ -^^ • J^7(i7+T=^ ^^^ '^ ] 



from which we derive the usual form for declinations which 

 are not too large t =ysec 8. 



And for stars near the pole 



sin t = sinysec 8. 



If the object of the observations be to determine differences 

 of right ascension, Bcssel's formula is the most convenient; be- 

 cause in that case it is not required to use the constant ?m. For 

 absolute determinations of time, the formula of M. Hansen is 

 more advantageous. 



XLIII. Report 



