THE VALUES OF 4- [' e' 1 " dt . 263 



_2 



will be less than that half. But the third series (11) is not convergent, the numerators 

 of the successive fractions soon exceeding any value of 2t 2 that is likely to be used. 

 To meet this case, we have Laplace's continued fraction,* into which the series is con- 

 verted, and which becomes more convergent the higher the value of t. And this can 

 be used for either G or H. 



Laplace's Continued Fraction. 



9. When t >1'5 it becomes very laborious to compute values of H, and Laplace gave 



f° a e~ t2 f 1 13135 1 



the series for] ( dt.e' 1 ' = — \ 1 — —^ + -^ — ^g- 4- etc. > , the form of a continuedfraction, 



putting 2=2^— 



O or Ce-^4— 1 (13) 



1+ 



2q 



1 + 



Sq 

 1+ 



4q 

 1 + 



1 + etc, 



and this gives a series of common fractions alternately greater and less than the integral. 

 Mr Glaisher has used this in computing his table of the values of the other function, G, 

 from t = S to t = 4:'5. And for higher values of t the approximation of the successive 

 fractions is increasingly rapid. But at any stage the degree of approximation can be 

 estimated only by reducing two consecutive fractions to decimals. To attain a nearly 

 correct value too, with values of t under 3, the computation of a long series of fractions 

 of the form — 



1 1 l + 2q l + 5q l + 9q + 8q 2 l + 14g + 33 g 2 



etc. 



1' 1+q l + 3q l + 6q + 3 q *' l + i0q+15q 2 ' l + 15q + 45q 2 +15q 3 ' 



becomes tedious. This is obviated to a considerable extent, by determining once for 

 all the coefficients a\ b', c', etc., and a, b, c, etc., in the following expressions for the 

 numerator and denominator of the fraction when it involves high powers of a. Thus 

 we get two consecutive fractions of the form — (when n is even) — 



l+a'q+b'q 2 + c'q 3 + . ■ . .+ l'q hn ' 1 

 ^- 1== l+aq + lq 2 + cq>+ . . ,+lq^ 1 



l + (a' + n)q + . . . + 1"^-' 



and L, = : ; : n 



^ l + (a + n)q + . . . + mq* 



md the numerator and denominator for L„ +1 are found by multiplying those of L„_ I 

 )y nq and adding those of L„. 



* See Laplace's M4c. CJl., ut sup., and Theor. Anal, des Probab., p. 104 ; De Morgan, "Theory of Probabilities, 

 63 ; and Biff, and Meg. Gale, p. 591. 



