(36) 



THE VALUES OF -MV**fife. 273 



through the rest, if he had not checked his work by means of Laplace's continued 



fraction. 



18. But the formula may be applied with great effect in this way : r may be taken 

 as negative as well as positive, so that from a value H, corresponding to t, we can derive 

 both the values at t-r and t + r; and by developing the formula more fully, we may 

 use it with much larger values than r = 0*01. Putting x= - 2t, the general term is — 



iT+I \ n~ (^^\ + 172X^^)1 3!(%-6)! + 4!(%-8)! j 



2 {' 

 Expanding and adapting to the integral —r e~' 2 dt, — 



att 2e-'-' (\ . 2( 2 -l, 2^-3^3,4^-12^ + 3, 4t°-20t s +15t . 

 AH = 7?r r {1-^4—3- r- g-r»+ ^ -*•* 3^ r« 



8* 6 -60* 4 +90* 2 -15 ,, 8* 7 -84* 5 + 210* 3 -105* . , 16* 8 -224* 6 +840**-840* 2 + 105 „ 

 3.5.7.3! 3.5.7.4! T 3.5.7.9.4! 



_ 16t 9 - 288l 7 + 1512ft- 2520* 3 + 945* , 9 32* 10 -720* 8 + 5040* 6 - 12600**+ 9450* 2 - 945 , 10 

 3.5.7.9.5! ? ' + 3.5.7.9.11.5! 



32* u -880* 9 +7920* 7 -27720* 5 + 34650**- 10395 * u 



3.5.7.9.11.6! r 



64 * 12 -2112* 10 +23760* 8 -110 880* 6 +207 900* 4 - 124 740* 2 + 10395 12 

 3.5.7.9.11.13.6 ! 

 _ 64* 13 -2496* 11 + 34320* 9 -205 920* 7 +54Q540* 5 -540 540* 3 +135 135* 13 j 



3.5.7.9.11.13.7 ! 'J 



For any portion of the table then, say from £=1*9 to t—Z, we may compute the 

 coefficients of the powers of r for t at the values 2*0, 2*2, 2*4, 2*6, 2'8, and 3 ; and by 

 means of the first we find the differences from f = 1*90 to £ = 2*10, by the second series 

 from £ = 2*10 to 2*30, and so on. If, also, we know the values for t = 2 and £ = 3 (which 

 I have computed separately, both by the general series and by Laplace's fraction), we 

 can fill up the table, — first, for all values of t differing by 0*01 ; and, secondly, by form- 

 ing from these values the differences in the series H + nA / + n ^zl A' 2 + etc., for 1/5, 1/10, 

 or any other subdivision of the interval, we may complete the table from £=1*900 to 

 t = 3*100. This sufficiently explains the method of computation for the portion of the 

 table beyond t= 1*000. 



19. Since the computation of these coefficients of the powers of r is also required 

 for the other branch of the integral — G, they may be preserved here. 



For t= 1, e" 1 = 0*367 879 441 171 442 321 595 524, 



AG=-re-'^l-r + ir 2 + L- 3 -^r 4 + ^r 5 + -036 5079r 6 -*011507 936V T -*004 541446 208112 875r 8 

 \ 6 6 90 



+ •002 954 144 620 811 287r 9 +*000 206 028 539 362r 10 - *000 481 935 7597?* 11 



+•000 045 088 6562r 12 + *000 057 110 732r 13 , etc. Y 



Also 4- e- l = 0*415 107 497 420 594 703 340 268 = E, and H = 0*842 700 792 949 714 869 34. 



* If we make <=0 in this series, r then becomes t, and we have the series in (9) from which it is derived. 

 VOL. XXXIX. PART II. (NO. 9). 2 T 



