THE VALUES OF -f- [e'^dt. 265 



~iyo 



_2_ 

 s/tt\ 



2 

 H = l-- f - : W-L = l-- 



Jir 2t tjic' (16) 



with L 23 we have H = 0-986 671 671 161 + 



and with L 2i we have H = 0-986 671 671 255 - 



the true value being, H = 0-986 671 671 219. 



Hence this degree of approximation, being to the tenth place in decimals, would be 

 practically sufficient for all purposes. And for higher values of t, the results are still 

 more close, and even a lower order of the fraction L would suffice. For t = 3, L 23 comes 

 out -951 813 839 1839 + , which is correct to the last — the 13th — figure. 



10. When the values of L„_ x and L n are not sufficiently accordant, either from t 

 being small or n not sufficiently high, we may readily compute L„ +1 . Then if L n ^ 1 -L n 

 = a, and L n -L n+1 = 6, we may find a correction ^ , ^, or -— (regard being had to 

 the signs of a and b, one of which is always negative), and — 



a 2 T ab r b 2 



L n -i + 7, or L„ + - — t> or L n+1 +- — ,, — which will be equal or very nearly so, — 



will give a closer approximation to the value of L than before. It will be greater or 

 less than the true value, according as ~L n . x and L, l+1 are both greater or both less than L, r * 



11. By means of equation (9) we may compute any values of H up to a certain 

 point with considerable facility, but with t > 1 it becomes rapidly more difficult. We 

 may, however, use it for such values of t as 2, 2*5, and even 3, though the work is 

 lengthy ; and for purposes of verification this has been done in the following table. 

 For extreme accuracy the continued fraction is scarcely less laborious, till we reach 

 ( = 3. Up to £=1"25 the values were determined for moderate and equal intervals by 

 means of (9), and the intermediate values inserted by interpolation, using the highest 

 order of differences that could by any chance affect the results. 



12. We might, however, make use of the method of quadratures. For H may be 



o 



regarded as the area of a curve of which the equation is y = T7^. e • Hence the value of 



^-e"' 2 represents the rate of increment of that area at t ; and the area between any two 



ordinates is the difference of the values of H between the two corresponding values of t. 

 A.nd if the intervals between the ordinates are so small as to enable us to find the area 

 with sufficient accuracy, we may compute values of H, — or rather of the differences of 

 H between two values of t, — with great precision. If, for example, we take the ordi- 

 lates, given in the first part of the table, from t= 1-160 to t= 1*170 inclusive, the area 

 s found by Simpson's rulet to be '002 904 196 086 + , and adding this to the value of 

 1 for t= T160 (from the second part of the table), the sum is the value of H when 

 = 1-170, viz., 0*902 000 398 966, — which is correct to the last figure. 



Or, generally, if V , V l5 V 2 , . . . V„, be the values of the successive ordinates whose 



* In the example above of t = 1-75, L 25 will be 0-883 925 237 509, and a= -6231, 6= +3854, whence the correc- 

 onB are, -3850, +2381, and -1473, respectively, each giving -883925 236 036. 



+ T. Simpson's Mathematical Dissertations (1743), pp. 109 f. This rule gives a very close approximation. Conf. 

 [ymers' Int. Calc., p. 181 ; Hutton's Mensuration, p. 374. 



VOL. XXXIX. PART II. W 9). 2 S 



