10 KEPORT— 1873. 



aud for Cape Town, or the position as under 3, according to what we have 

 derived from all local English observatory journals, by 



d= 29 33-85-6-11273 (<-58-04)=, 

 i=_58 51-07 + 0-02242 («-165-58)S 

 w=588-9o + 0-02813 ((-61-806f. 



Now by developing out of each of these expressions, with ^=435, their 

 absolute values as well as the annual increments cd, H, and lu) of the same, 

 and then introducing these quantities into the easily proved expressions, 



2X=cos d . (jw — (0 . sin 1' . sin d . Id, 



SY=sin d! . 2w + w . sin 1' . cos d . hd, 



2Z=tan i, hu) + w . sin 1' . see' i . hi, 



these increments of rectangular components for 1843-5 are obtained as above 

 under 1 and 3. 



But for all the other above-named places, the existing observations, when 

 treated as the last mentioned, did not give complete expressions for d, i, and w, 

 but only their expressions for limited periods. The annual increments of the 

 components X, Y, Z, which were determined from such observations, in 

 general did not exactly pertain to 1843-5, but to a value of t somewhat 

 different from 43-5. Now, as our computation for the first epoch, or 1811, 

 had already furnished the increments of the constants a,, a^. . . .a^ for the 

 same, we have, first, calculated (by the help of the following formulae (3), 

 (4), and (5)) the annual increments of X, Y, Z at the same places for 1811, 

 and then, having denoted the value of any one of these increments for 



1843-5 by S, 



1811 by \^, 

 1800 + < by St, 



we have determined the results, as given above under number 2 and 

 numbers 4 to 13, by the relation 



^=S. + |^. (43-5-0. 



There were, in particular, to bo used for the increments under numbers 



2, t=41-4, 

 4, « = 35, 

 5 and 6, t=48-5, 

 7, t=38-o, 



12, t=3G, 



13, «=45; 



whereby it appears that the empirical elements of our equations were in- 

 fluenced, to an always slight but not wholly equal extent, by a former calcula- 



