260 DR JAS. BURGESS ON 



immediately from the table for the integral fe~ tl dt in Bessel's Fundamenta Astronomies* 



There seems to be a mistake here, for the table could be derived directly only from 

 Kjramp's Table I. 



De Morgan reproduced this table also in his " Theory of Probabilities" (Encyclop. 

 Metroj)., 1837), and again in his Essay on Probabilities (1838), but there he extended 

 it to £ = 3*00 from Kramp's data. Again, Galloway, in his " Treatise on Probability" 

 (1839), prepared for the 7th edition of the Encyclopaedia Britannica, printed Encke's 

 Table, also continued to the same point. 



6. Further, and in dependence upon this integral, Encke gave a table t of the values of 



2 C 9 ' 



'"■dt = K, (5) 



7^1 £ ' 



J 



p being the numerical value of t when H = J, giving 0'5 for the value of the integral K 

 when the argument is T ( = pi) = 1. His table gives the values of K to five decimal places 

 only with the argument T, at intervals of 0-01 from T = to T = 3*40 and at intervals of 

 1 from T = 3*4 to T = 5. It was computed from the previous table by direct inter- 

 polation, and was also reprinted by De Morgan both in his Theory and his Essay. 



Here it may be noted that this second table is so readily derived from a table of the 

 values of H, when these are determined with precision, that there seems little reason for 

 computing it. For if we multiply the arguments in such a table by l/p = 2*096 716 165, 



or approximately by ^ or oaq, we have at once a table of the values of K, only with 



arguments at intervals that are inconvenient on account of the fractions. But since the 

 arguments required in practical applications nearly always lie between two consecutive 

 tabular arguments, and interpolation has to be made at any rate, we may as well perform 

 the operation on the values in a table of H as in one of K. This is done by multiplying 



the argument (T) for K by p = 0*476 936, or, approximately by -^ 5 , and taking the corre- 

 sponding value from the table for H. Thus, if for the argument for K we have T = 372, 

 then 372 xp- 17742 = t, for which our table gives H = 0*987 8960 : and Encke's table, 

 by interpolation, for arg. 3*72, gives K = 0*98790. 



But, we might also compute the first part of Encke's table from the formula — 



K = 0-538 164 958 101 235T--040 805 140 181 145T 3 +*002 784 561 677 8354T 5 

 -•000 150 809 348 77027T 7 + -000 006 670 286 943 3025T 9 -*000 000 248 189 408T" 

 + -000 000 007 964 597 724T 13 - -000 000 000 224 304 823T 15 + -000 000 000 005 627 456T 7 

 -•000 000 000 000 127 2874T l9 + etc. (6) 



This will give values correct to fourteen decimal places, as far as T=l, and seven 



* Berl. Astronom. Jahrbuch fur 1834, S. 269. Mr J. W. L. Glaisher (Phil. Mag. (1871), vol. xlii, p. 434) remarks 

 that, if Encke's table were derived from Bessel's, it must have been "by interpolation from his second tabli 

 But he overlooks the fact that Bessel's Table II. is only a continuation of Table I., giving the logarithmic values ol 

 the multiple of the integral by e ( from t — 1 to <=10, with logarithms of t for argument. 



t Berl. Astron. Jahrb., 1834, Ss. 309-312. 



