and the Methods of calculating their Results. 343 



this supposition the values of P and Q corresponding to the 

 time T + T', are obtained by this formula : 



(7)... g + T.ft + — . C + fl>3 .</+—-- — e+kc. 

 consequently, p' and q' by this formula : 



by which N and log n are found. If, however, several ob- 

 servations are to be calculated at the same time, it is more 

 convenient at once to calculate the values of N and log n for 

 different values of T'; they are obtained by putting for n sin 

 N and n cos N these expressions : 



For T= - 2 b -c + \d- 



1 b'~\c 



b — £ d 



_ i 



2 «* 4 



== + 1 b + \c 



= -f 2 b + c i- \ d f ±e. 



For every single observation M and log m are then to be 

 calculated by the formulae 



7>i . sin M = p — u, m cos M = q — v 



in which p and q are the known values of P and Q for the 

 time T; at last the sixth formula is to be calculated. A se- 

 cond approximation is not necessary if the value of T' = 

 t — d — T, or the assumed difference of meridians employed 

 in the first is correct within some minutes of time, which may 

 always be supposed. Such an approximation would, how- 

 ever, cause little trouble, as the calculation of formu1a(6), only, 

 would be affected by the alterations. A second way of per- 

 forming the calculation would be to calculate by interpolation 

 (7) from the values of P and Q corresponding to T, T + 1, 

 &c. those values of these quantities which belong to the time 

 of observation, reduced to the first meridian by an assumed 

 difference of meridians, and to use these values instead of 

 p and q and their differential quotients, instead of p' and q', 

 The latter are found by the formula 



O TV 2 1 O T'3 T" 



(9) ... b + T.c + —~ d + \ A e + &c. 



by the application of which, consequently, N and log n will 

 be obtained. If it should be preferred at once to find N and 

 log n, and to interpolate between the values thus calculated, 

 they might be obtained by the expression for n sin N and n 

 cos N, 



For 



