352 Some New Mechanical Quadratures. 



Bertrand, Gauss's formula for ?i = 4 gives a result which is 

 too small by 3 units in the seventh place, and thus tested it 

 is intermediate in accuracy between (5) and (6) *. 



As^ an application of the formulas here developed I may 

 mention the double integral of the probability curve. By 

 any of the more accurate formula (2) to (7) it may be shown 

 that 



\ djc I e a x — - . 

 J o J l 



In computing it 7-place values from Burgesses table were 

 taken and the result obtained coincided with I/x/tt accurately 

 to 7 places. This value is of interest in the analysis of 

 diffusion. 



Values of the probability integral itself are commonly 

 arrived at by a somewhat intricate process, better fitted to 

 yield a related set of values than a single one. By the 

 formulas given in this paper applied to tables of e~ x isolated 

 values of the probability integral are readily determined f . 



* Rather curiously Weddle's rule applied to Bertrand's problem gives 

 somewhat better results than (3). Furthermore, as this rule is deduced 

 from Newton's interpolation formula it appears to err only by a small frac- 

 tion of the sixth dirt erence, when the seventh difference is negligible. As 

 here deduced from Euler's equation the error should include fifth 

 differences. While these facts are not incompatible the relations seem 

 to need confirmation, and I have integrated e x dx from x=. —1 to x=2'6 

 taking h = 0'6, and using values of e x with 7 significant figures. By 

 a separate computation I find the true value of the integral to be 

 13-09585 85938. Weddle's rule gives a value which is too great by 

 000064 while the value of the corrective term given in this paper for 

 his rule is 0-00073 or about 9/8 of the real error. Formula (3) gives a 

 value 0-00016 too great or 1/4 of the error of Weddle's rule and 2/3 as 

 great as the value of the derivative term in (3). 



t For .r=0'7, Burgess's table (Trans. R. S. Ed. vol. xxxix. 1900, p. 257, 

 gives a value of the probability integral greater by 2 in the seventh place 

 than that assigned to it in Encke's table (Ast. Jahrbuch, Berlin, fur 1834). 

 This is founded on Kramp's table (Analyse des refractions astronomique 

 et terrestres, Leipsic et Paris, an VII. [1799]) and has been adopted by 

 Airy, Kelvin, and others. A physicist not familiar with the history of 

 these tables might wish to ascertain which value is correct, and this may 

 be accomplished by the help of (5). With /?. = 007, seven-place values of 

 y — e-' x ' x may be taken out of Smithsonian Math. Tables. Integrating by 

 (5) and multiplying by 2/yV gives the required integral at 0*6778012 

 which is Burgess's value. If a computing machine is available, the 

 arithmetical work is no more extended than in the example worked out 

 in a previous footnote, and it requires no mathematical knowledge beyond 

 that required for interpolation to one additional place in the table of the 

 exponential, so that a school-boy can do this " sum " on a single page of 

 note-paper. 



