278 DJK JAS. BURGESS ON 



JL e -' s rf< = 0-498 258 053711 787 564127 43 = H. 

 JttJ 



For the same value of t, we have— 



JL € - (2 = 0-900 466 098 615 398 685 314 176 = e , 



and Kramp's formula (36) for a difference r, becomes — 



i-H = er(l-0-475r -0182 916r 2 + 0-201 776 0416r 3 + 0-016 537 552r 4 

 - 0-056 42539r 5 + 0-00372r 6 +etc.) 

 = 0-900 466 098 615 398 685 314 176r- 0-427 721 396 842 314 8755r 2 

 -0164 710 257 205 0667^ + 0181 692 485 0336r 4 + 0-014 891 505r 5 

 -0-050 809r 6 +0-00335r 7 + etc. 



Using the first three terms, and taking r = '001 936 as a first approximation, we 

 obtain H'-H= + -001 741 699 03, etc. But ±-H = '001 741 946 29, and the differ- 

 ence of these is i-H'= +'00000024726. The value of fa'* for £ = 0'476 936 is 

 0'898 80814. Hence the correction is " 00 fi 2 Q ^ 26 = + 0'OOQ 000 275, and the new value 



of p is 0'476 936 + '000 000 275 = '476 936 275 ; and from this,— taking in the higher 

 powers of r, — we readily arrive at the value, correct to the twenty-fourth place of 

 decimals, viz.: — 



p = 0476 936 276 204 469 873 383 506. 



Otherwise, we may form a difference-formula for the computation of this and other 

 values of t corresponding to definite values of H. Thus let H be the tabular or com- 

 puted value corresponding to t, and H' the value for which t' = t + At is sought. Put 

 h =£ «/7r(H' - H)e -'\ Then— 



w 7.1 , 7, , & 2 + h z , 12 * 3 + ^ z,3 i 96^ + 92^+7^ , 480*5 + 652*3 + 127^ , , x ,« fl v 

 At = h(l + ht-\ g — h 2 -\ g h 3 -\ 235 h + 35-g A 5 + eta). (38) 



Using the above value of H for £ = '475, we find h= '001 934 494 025 806 1229, and 

 this series becomes — 



At = h(l + -475h + -634 16h 2 + -768 510 416/V + 1-088 151 25A 4 + 1-575 641 961 805>+ . . .), 



and this gives at once the value of p correctly to seventeen figures. When h is very 

 small, the first three terms of (38) will usually be sufficient to determine the values of 

 t corresponding to H = 0'1, 0'2, 0'3, etc., as given below, § 23. 



21. The following table contains the values of the factors dependent on this con- 

 stant, p, together with some others used in Probabilities,* with their logarithms, com- 

 puted to a degree of accuracy far beyond what can be required. 



* These constants will be met with, among other places, in Bessel's Fundamenta Astron., p. 18 ; and Ueber d. 

 Bahn des Olbersch.cn Kometen, in Abh. d. Math. Kl. d. Kdnigl. Preuss. Akad., 1812-13, S. 142 ; De Morgan's Theory of 

 Probab., §§ 68, 100, 116, 150, 152, etc. ; Encke, in Berl. Ast. Jahrb., 1834, Ss. 270, 293, 298 ; Gauss, Werke, Bd. iv, S. 

 6 ; Aiky, Theory of Error s s pp. 23, 24 ; Poisson, Rech. sur la Probab. des Jugements, p. 176, etc. 



