324 REPORTS ON THE STATE OF SCIENCE, ETC. 



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The Probability Integral je-i'^dt and its Integrals. 



For the sake of convenience and simplicity the integral is denoted by Io(a;) and 



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those derived from it by repeated integration, by I„(a;) ; i.e. I„(a;)= I I,i_i(a;) dx. 



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As Dr. H. Jeffreys lias pointed out, the most commonly used notation is Er/x, but the 

 introduction of the symbol I„(a;) need not lead to any confusion with the Bessel function 

 with imaginary argument. 



(A). For small positive values of x the series in ascending powers of the variable 

 are convenient. 



\i\\x) = A / — — a;l— -.— + _. — - . + . . . 1 



' V2 V 2 345.2! 8 7. 31 / 



whilst for large values of x the asymptotic series can be used. 



Ux) = 'L^{\- l_|_l-3_1.3.5^ 1.3.5.7 _ 



a; \ a;^ x* sfi x^ 



The series in the bracket, where the signs of the terms alternate, is an asymptotic 

 series of the first kind (Stieltjes), and can therefore be employed to give results with 

 an error considerably smaller than the least term. As shown in the 1926 Report, 

 several places of decimals can be added to the result obtained when the divergent 

 terms of the series are neglected. Intermediate values of \(x) were obtained by 

 calculating first differences over smaller intervals, 0-1, from e~4'*, the differential 

 coefficient of the function, as in the case of the sine and cosine integrals. 

 Functions of higher order are found from the recurrence formula 



nl„ (a;) + xl„ _ i (a;) — 1„ _ 2 (a-") = 



where I_i («) = e-i-^'^ 



Owing to the accumulation of errors, the formula is not very suitable for large 

 positive values of x. The continued fraction 



In-2(-t )_^i n w+l w+2 n+3 

 \n—\{x) x-\- x-\- x+ x+ 



giving the ratio of two functions, has been applied in these cases. For example, 

 when x= 4, the ratio of Iio(x) to Iii(a;) is equal to 5-97. Ratios of lower order functions 

 were then computed, with the following results : 



p (10,ll)=5-97 

 p ( 9,10)=5-84 

 p ( 8,9) =5-711: 

 p ( 7,8) =5-576 

 p ( 6,7) =5-4348 

 p { 5,6) =5-2880 

 p ( 4,5) =5-13464: 

 p ( 3,4) =4-97377 7 

 p ( 2,3) =4-80421 78 

 p ( 1,2) =4-62445 129: 

 p ( 0,1) =4-43248 3742 

 p (_l,0)=4-22560 71445. 



The value of I_i{4)=0-0''3354626279 

 and of Io(4)=0-0^7938803027. 



The ratio of these two values agrees with the last ratio in the foregoing table to ten 

 places of decimals. 



