264 DR JAS. BURGESS ON 



Thus, putting L,= T \ L„ +1 = '" It> " ( 15 ) 



When >i is an even number the fraction L„ is less than the true value, and when 

 odd, it is in excess by a quantity c <^(L,~L„ +1 ). 



The larger t is, the more rapidly the fraction approaches its limit, and consequently 

 a lower value of n in L„ will give a sufficiently close approximation. 



The following values of the coefficients of q in L„ can be made to serve in nearly all 

 cases when £>1*5 : — 



L _ l + 77g + 20702 s + 23814g 3 + 114 765^+187 425g 5 + 46080g 6 



13 1 + 78g + 21452 2 + 257402 s + 135 135? 4 + 270 270? 5 + 135 135? 6 " 



r 1 + 90g + 2915? 2 + 42300? 3 + 278 01 V + 729 330? 5 + 509 985g 6 



14 ~ 1 + 91? + 3003? 2 + 45045? 3 + 315 315? 4 + 945 945?° + 945 945? B + 135 135? 7 + 



t 1 + 104? + 3993? s + 71280? 3 + 611 415? 4 + 2336 040^ + 3133 935? c + 645 120? 7 

 5 ~ 1 + 1052 + 4095? 2 + 75075? 3 +675 675? 4 + 2837 835? 5 + 4729 725? 6 + 2027 025? 7 " 



t _ 1 + 135? + 70072 s + 178 893g 3 + 2386 395? 4 + 16288 965?° + 51450 525? 6 + 58437 855y 7 + 10321 920? 8 _ 

 17 " 1 + 136? + 71402- + 185 640? 3 + 2552 550? 4 + 18378 360? 5 + 64324 260? 6 + 91891 800? 7 + 34459 425? 8 



L<ifi=- 



1 + 152? + 9030? 2 + 269 724? 3 + 4341 480? 4 + 37469 520? 5 + 162 058 050? 6 + 297 693 900? 7 + 151 335 135? 8 



8= "l + 153? + 9180? 2 + 278 460? 3 + 4594 590? 4 + 41351 310? 5 + 192 972 780?° + 413 513 100? 7 + 310 134 825? 8 + 34459 425?' x 



1 + 189? + 14348? 2 + 567 420? 3 + 12686 310? 4 + 162 912 750? 5 + 1167 180 300? 6 + 4302 906 300? 7 + 6859 400 625? 8 J 



j +3061 162 125?" i 



-" = 1+T90? + 14535? 2 + 581 400? 3 + 13226 850? 4 + 174 594 420? 5 + 1309 458 150? 6 + 5237 832 600? 7 + 9820 936 125? 8 ) 



+ 6547 290 750? 9 + 654 729 075? 1 " ( 



1 + 209? + 17748? 2 + 796 620? 3 + 20603 310? 4 + 314 143 830? 5 + 2775 672 900? 6 + 13408 094 700? 7 + 31335 467 625? 8 \ 



r +27125492625 g 9 + 37158912007 1Q J 



21 = 1 + 2102 + 17955? 2 + 813 960? 4 + 21366 450? 4 + 333 316 620? 5 + 3055 402 350? 6 + 15713 497 800? 7 + 41247 931 725? 8 \ ~ 



+ 45831 035 2502 9 + 13749 310 575? 10 f 



1 + 252? + 263152 s + 1488 384? 3 + 50044 7702 4 + 1033 829 160? 5 + 13108 004 910? 6 + 98983 684 8OO2 7 + 416 674 583 325? 8 \ 



T = + 860 553 193 500? 9 + 664 761133 575? 10 +81749 606 400 ?" J. 



2! 1 + 253? + 265652 s + 1514 205? 3 + 51482 970? 4 + 1081142 370? s + 14054 850 810?" + llu 430 970 650? 7 + 496 939 367 925g 8 I 



+ 1159 525 191 825?' + 1159 525 191 8252 10 + 316 234 143 225?" I 



1 + 275? + 316052 2 + 1987 8752 3 + 75297 1142 4 + 1781 769 150?s + 26460 800 7302 6 + 241 511 019 7502 7 + 1288 808 846 325r/ 1 



r + 3659 572 691 775? 9 4 4601 737 965 825? 10 + 1645 75 6 410 375?" / 



- 1 = 1+ 2762 + 318782 2 + 2018 9402 3 + 77224 455? 4 + 1853 386 9202 5 + 28109 701 6202 B + 265 034 329 560? 7 + 1490 818 10:3 77;Yl" 



+ 4638 100 767 3002 9 + 6957 151 150 9502 10 + 3794 809 718 7002 n + 316 234 143225?" I 



The multiplier q being always a proper fraction, we begin by dividing the last 

 coefficient by 2t 2 , add the next preceding and divide again, and so on to the 

 first coefficient of q, adding unity to the last quotient. If, for example, we take 

 t=l'75, 9 — ^=^ + 42' — which is easily manipulated — and we find, on dividing down 



the coefficients in the terms for L 23 — 



1 007439-089305 

 1139733-366404 



L^ = ™„£* n ™l A " A = 0-883 925 239 886, 



. T 2535470688789 n , Mq9 „„ u ,, 



and L * = 2868422-115642 = ° 883 ™ 233 655 ' 



These agree to the eighth decimal place, the first being too large and the second too 

 small but nearer the true value, — which is 0883 925 236 007 66. 



For t=V75, the value of e^ is 0052 774 995 930 150 374 66, and since (4)— 



