272 DR J AS. BURGESS ON 



A' = A' + 2 t ?i A?, A:' = A 3 — 2 ~'"a\ A»=A 2 +4^A s e , A e =A--^A?, and V„=V+nA e 



O 4 6 A 



To bisect an interval, n=£-, and-— 



v _ V ,A A 2 _A 8 3A* 3Aj 5A" 5A 7 35A S 

 * + 2 8 16 + 128 + 256 1024 2048 + 32768 + etC> (33) 



Or, A*=A 4 +KA 5 +£A 6 ), A; ! = A :! -gAJ, A? = A 2 +£A?, A c = A-|A?, andV, = V+|A c . (34) 



Thus if it be required to find the value of H corresponding to t— 1 '575, we take the 

 differences following and on line with 1"574 in the table, and proceed thus : — 



A H 



+ 188 870 390 940 -973 983 952 882 675 

 = +297 470 570 . = + 94 583 930 75.' 

 -1S669X -f 1 + 5992293x£' - 1 189 882 281 x - V +189 167 861 510 x £ "974078 536 813430: 



This value, H = '974 078 536 813 430, is correct to the last figure, and ^-A 5 = - 13, is so 

 small that it might have been neglected without affecting the result. 



After determining the values of H for moderate intervals, the differences for the 

 smaller intervals of '001 or '002 were determined by means of the formulse (22) to (25), 

 and the table thus filled up throughout. 



The Difference Formula, 



17. The difficulty of computation, due to the slowness of convergence of the series 

 for values of t above 1*0, led Kramp, who computed the table so often reprinted, 

 to adopt a difference-formula* obtained from the general series by means of Taylor's 

 theorem, viz. — 



A 4 A 3 



A 2 



-18656 +5985 292 



-1192 978 427 



-13=4A 5 ,= +7 001 



i= +2 996146 



k f e -edt=-r e -<il-rt+ 2 -^r*-^^r* + ete\ 



(35) 



where A^ = r = 0'01. This implies the separate computation of the values of the differ- 

 ences for each entry in the table. When r is small, three terms of the series maj 

 sufficient, and M. Kramp says he used no more. Mr J. W. L. Glaisher, in computing 

 the values of the same function from £ = 3 to t = 4'50, tells us that he computed separate 



tables of loo;. e~ r and of loo;. ( r — tr 2 + — r 3 — V 4 ), and then built up his table 



by the successive differences. t This requires for his table about a hundred and fifty 

 computations of the values of (35), and an error in one would have been perpetuated 



* Analyse d !' 'ra i\ ns astronom-iques et terrestres (Strasbourg, 1799), p. 135. 



t Philos. Mag., xlii, (1871), p. 434. Conf. De Moroan, ut cit., § 117. Mr Glaisher remarks (p. 432) that 

 " Kramp does not state what value he started from in applying the differences, or what means of verification he adopted. 

 Id all cases where a table is constructed by means of differences, the last value should be calculated independently 

 and then the agreement of the two values would verify all the preceding portion of the table." And he adds that 

 Kkamp's value For '=3 is in error in the tenth and eleventh figures, so that probably a portion of his table is incorrect 

 in the Last two figures (see § 4 above). 



