8 C. V. L. Charlier. 



the theory of probabihty, as the standard (mean) deviation says all that is wanted 

 from the calculus in the respect that here is concerned. 



The values of the probability function tp {x) are most conveniently tabulated by 

 Sheppard (»Biometrica» 1903). The argument of these tables are the quotient (,/• — b): n. 

 In the same memoir also the values of the probability integral, that is of the integral 



j'^ (a:) <lx 



are given in a similar manner. 



As to the form of the derivated functions of I remind of the relation 



f (.«) = J?. (./:)?(*:), 



where i?, {x) is a whole rational function (i. e. a polynom) of x of the degree s. 

 For the lowest values of s we have 





1, 











(x-h). 





11-,=^ 



+ 



{■r-bf - a^ 





0« R, = 







r-b), 



^- -K, = 



+ 



{x—b)'^ Gi-(, 





^.^^R,= 











4- 



(X^b)''— 15 G-(. 



x—b)' + 30 [x—by 



Hence we find that o' R, is a function only dependent on the "quotient 

 (/; — b): a. As the product '^'{•{x) also depends only on the same quotient, it is obvious 

 that the functions 



^ x{x), a-^'f"' (./■), a'''f'^- (,/■), (■'■). ■ ■ • 



are functions only of a single variable and hence may be conveniently tabulated 

 with this variable as tabular argument. 



I give a short table of the first three of these functions as well as of the 

 probability integral at the end of this memoir. 



In many instances the following abridged table will suffice for constructing 

 a frequency curve (compare (5*)): 



TABLE I. 



X — b 





^0 





f4 



— 3.0 



+ 0.004 



+ 0.080 



+ 0.133 



-2,5 



+ 0.018 



+ 0.142 



+ O.UiO 



— 2.0 



+ O.U5t 



+ O.IOS 



— 0.270 



— 1.5 



+ 0.130 



— 0.146 



— 0.704 



— 1.0 



+ 0.242 



— 0.484 



— 0.484 



— 0.5 



+ 0.352 



— 0.484 



+ 0.550 



0.0 



+ 0.399 



0.000 



+ 1.197 



+ 0.5 



+ 0.352 



+ 0.484 



+ 0.550 



+ 1.0 



+ 0.242 



+ 0.484 



— 0.484 



+ 1.5 



+ 0.130 



+ 0.146 



0.704 



+ 2.0 



+ 0.054 



— O.108 



— 0.270 



+ 2.5 



+ 0.018 



— 0.142 



+ 0.080 



+ 3.0 



+ 0.004 



— 0.080 



+ 0.133 



