372 Astronomical and Nautical Collections. 



.000261 30+.00001610 + .00000359 + .00000177 + [.000 002 40J 

 = .00028516, too much by about .00000164, and we must sub- 

 tract -ji^, and we have r — .002586 = 8' 5".34, with an uncer- 

 tainty that cannot exceed a few seconds : Mr. Ivory's table, which 

 may possibly be correct, but which would naturally be a little 

 within the truth rather than beyond it, since it is computed by a 

 direct converging series, has 8' 48".0, which is 5".4 less. It can- 

 not be supposed that Mr. Ivory's method requires any such con- 

 firmation, but it would be easy to add a few more terms to this 

 series as a test, if there were any necessity for the perfect accuracy 

 of the determination by two opposite methods. 



C. The approximation lately communicated to the Royal Society, 



which supposes y = 1.5z'' — .5z^ gives ^ = -E t= 2.25 \/z — z, 



dz 



whence ^ =: -L-.= i£i. _ JL, i^JL=zl:l^\ 

 dr mpsz mps ,J z nips dr mpszj^ z 



d-v 1.125u tv. /J • e ^i.- ■ -.• 1, 1.125 J 



^ — = — , the fluxion of this initially — dv — 



dr^ mp'S'ZA/ z mp"s^ 



.6875u , , dH 1.125 dv , 1.6875 v' 



dz, and — — = __ -f — : cOnse- 



2 dr^ mp'-s- dr mp^s" ps 



quentlyp = iL r + (2.85275 - Lss)— + 3019 w iL + 755 

 s 2 ss s^ 



(4.7055 + 5291^2) !l: and, for r = .0026, p ^ .0026130 + 



s* 



.000016 lO+.OOO 005 39 + .000 002 03+[.000002 00.] = .00028682, 

 requiring for r a reduction of -y^^, whence r ~ 0025740 = 8' 50".9, 

 with an uncertainty not exceeding 2'' at the utmost. And the 

 direct computation by logarithms gives 8' 49".6, differing only 1".3 

 from the series in this almost extreme case : the series being in 

 this hypothesis a little more rapidly convergent than in some 

 others, so that it would be unnecessary to compute more terms if 

 it were to be employed for any practical purpose. It may be 

 remarked that the omission of v^ in the fourth term is not quite so 

 unimportant to the result as it appeared at first sight, though it is 



