Mr. J. W. L. Glaisher on a Class of Definite Integrals. 435 



fraction and foun d = '000 000 0] 3 663 1 89 ... ; the differences 

 corresponding to argument between 4*0 and 3*5 were then added, 

 and Erf 3*5 was obtained ='000 000 658 553 74..., agreeing 

 with the value found by direct calculation for Erf 3*5 as far as the 

 thirteenth figure. The values from x=35 to #=4'0 were there- 

 fore entered in the Table to thirteen decimal places ; and it is not 

 probable that any is in error by more than a unit in the last place 

 in any of the values. Starting again from 4*0, the differences cor- 

 responding to arguments between 4*0 and 4*5 were subtracted, 

 and Erf 4-5 was thus obtained = -000 000 000 174 250 . . . By 

 direct calculation from the continued fraction Erf 4*5 was found 

 = ;000 000 000174237 6... As these values only differ by 

 unity in the fourteenth place, the portion of the Table from 

 x =4*0 to # = 4*5 was given to fourteen places, it being un- 

 derstood that the last figure may be in error to the extent of one 

 or even two units. 



From the above description it will be seen that the mode of 

 verification adopted formed a very perfect test of the accuracy of 

 the Table. I rather regret now that I did not tabulate e x2 Erf x 

 in preference to Erf x. When a function whose value is numeri- 

 cally small enters into analysis, the term involving it can usually 

 be neglected, unless it contains also a very large factor, which is 

 usually an exponential. It is also a matter of observation that 

 when a function is expansible in a descending series multiplied 

 by an exponential, this factor points out the factor which it will 

 be convenient to multiply the function by previously to tabulation; 

 thus from (55) we see that e^Erf^ is a function which only 

 becomes infinitely small for very large values of x. Similarly 

 when x is large, it is preferable to tabulate e _cc Ei x and e x Ei(— x) 

 in place of Ei x and Ei (— x). This deficiency in the case of the 

 error-function I hope to supply by forming a Table of e x2 Erf a? 

 for arguments above 3. 



Since Erfc x—\ ^tt— Erfa?, the value of JVtt to fourteen 

 places of decimals is printed on the same page as the Tables, to 

 facilitate the deduction of Erfc x from Erf x. 



2F2 



