356 Astronomical and Nautical Collections. 



a' being the variation of the declination from the time of the 

 afternoon until the time of the observation of the following fore- 

 noon, and D' the declination for midnight, or the time equi- 

 distant from both observations ; calling the interval in time T, 



we have as before 



T 



x' — —. djD 

 24 



{?• + P")- 15° (24— T') 

 H^'+0= 180° - (15-|.'). 



Substituting these values in the expression of ^ (P" — P), 

 keeping in mind that 



Sine (180° - M) = Sine M 

 Tang. (180° - M) = — Tang. M, 

 and dividing by 15 to reduce it to time, it follows that 

 J,(P" — F) _ \T 



15 .,^„ e- ,,.o T 



Sin- 



r 



dDTang. L 



360 Sine (15° I.\ 



d D Tano-. D 



360 Tang. (1 5° 1^") 



:= ^'dZ) Tang. Z, + B'dD Tang. D' 

 Finally substituting in the equation (jE) 

 Time of apparent midnight by the chronometer = \ {H' + i7") 



+ .4dDTang. L 

 + Bdi)Tang.X)' 

 Comparing this expression with that which we have before 

 found for noon, we see that they only differ in the sign of the 

 first term of the correction. And thus, taking altitudes of the 

 sun in the afternoon, and corresponding ones the following 

 forenoon, we can find the time of midnight by the chronometer 

 with equal facility, and by the same calculation as for noon, 

 only changing the sign of the first term of the correction. 



The interval T or T ought strictly to be computed in apparent 

 time ; but good watches go so accurately, that 24 hours by the 

 watch will be so nearly 24 hours of apparent time, that the in- 

 terval shown by the watch may be considered as the interval of 

 apparent time, without the leqist risque of any sensible error. 



