September 2, 1921] 



SCIENCE 



201 



is laborious, excessively so -where, as in the 

 present case, many decimal places are requir- 

 ed, it is possible to make a closely approx- 

 imate integration of small segments of the 

 function : 



ydx = e-^-dx 

 such as 



/: 



-(.a + zr-dz. 



where a is any abscissa and z is small. 

 Expanding the exponent, this becomes: 



/:. 



which by putting 

 becomes 



e-'-= 1 — 2', 



_,[-,, / z'-l z J^\ 

 - * L^ '"' I 2a '^ 2a' + 4aV 



-«"''(^^-2^+4?j J- 



Eeducing and substituting for z the value 

 1/10 gives: 



I g_(a + j)^ (e«;5 + g-a'5)(j/5 



J_i = 4a' L 



+ (ec/6_e-a;5) (l.98o2 — 1) 1 



Thus, by assigning values to a, progressing 

 by 0.2, the areas of the segments of the inte- 

 gral for the abscissal intervals a ± 1/10 

 could be closely approximated and summated, 

 the values in the table being finally: 



« (2 r^e-'^'-^-dx): 



loi 



or log (1 — P), according to the usual sym- 

 bology. It was found that it was only neces- 

 sary in the extreme value given (hx ^ Y.O) 

 to carry the computation a few steps farther, 

 in order that the sum of the subsequent seg- 

 ments to infinity should be a vanishing quan- 

 tity with respect to the degree of precision 

 decided upon. The table is not to be looked 

 uiwn as more than supplementary to the 

 tables in general use, and upon examinatioii. 



it will appear that the error introduced by 

 assuming that e"^° = 1 — z^ is negligible 

 since, for z = 1/10 this error at its maximum 

 is only as 0.99 — 0.99005 to 0.99, or 5 parts in 

 99,000 with respect to 1 — P, and on the whole, 

 even less than this; and it is the values of 

 1 — P, smaller than those obtainable from the 

 usual tables, in which we are here interested. 

 The values of this table check with those in 

 the usual tables, as far as the latter go, and 

 also (in the extreme cases, especially where 

 hx = 5.0, 5.5 and 6.0) with the values given 

 in the original work of Burgess.^ 



EXPLANATION OF TABLE 



Common logarithms of the values of the 

 integral : 



2:^£e-'^'x"-dxi=l-P) 

 for various values of hx. 



kX' 



0.4769x _ 0.7071X 

 ' E ~ ,T ' 



where E is the probable error and »■ the quad- 

 ratic mean error. 



Interpolations will be fairly accurate to the 

 fourth place if proper account be taken of the 

 second difference. 



hx log (1 — P) 



0.0 0.0000 



0.1 9.9482-10 



0.2 9.8906 



0.3 9.8270 



0.4 9.7571 



0.5 9.6808 



0.6 9.5978 



0.7 9.5081 



0.8 9.4115 



0.9 9.3077 



1.0 9.1967 



1.1 9.0784 



1.2 8.9527 



1.3 8.8195 



1.4.... 8.6787 

 1.5.... 8.5301 

 1.6 8.3739 



3 Burgess, Trans. Boy. Soc. Edini., XXXIX., 

 p. 257 ff. "On the Definite Integral (Z/v) f„'e-^-dt 

 with Extended Tables of Values." 



