Researches into the theory of proljaliility. 



29 



where 



(2S) I- 



^ ' (0* |4 = V, — 3v2 + 5(0^7, — 6(ov,. 



These expressions we may also write in tlie foUowiug form 



^^-13 ^^"13^^'' 



|4a^ ^ +24 



or, introducing the coefficients and ß, belonging to the curves of type 

 (29) 



T4 = ^^ + |-ß3>-^ + ß. 



in which form the calculation of the coefficients for the curves of type B is easily 

 perforuied. 



For graphical construction it will be suitable to write the equation of the 

 frequency curve in the form 



(30) f .F(.;co + c) = l ^r['H.r) + Y,,A-^i. + Y,A^'l. + . . .]. • 



The fornuikc (25), (27) and (28) contain all that is necessary for the calculation 

 of the curves of type B. The iiumerical operation is substantially the same for 

 the curves of both types. The calculation of [x^,, v^, v,, v^, a, ß.,, ß^ is executed 

 according to the scheme II. Then X and co are calculated with the help of (25), 

 and -/-.J and b}^ the formulœ (29). The graphical construction and the cojuparison 

 with the ohservation is performed with the help of (30). As for the present the 

 values of the function <[{x) are tabulated only for 'integer values of the argument 

 the comparison between observation and theory must take place in graphical 

 manner. The values of 'j)(.r) for integer values of x are given according to Bortke- 

 wiTSCH, in tab. E. 



It is supposed in these investigations on the curves of type J5, that 



(3 1 ) to i: [xo, + cy F{xM + c) = x- f{x\ 



where x takes all integer values between • — co and -|- co . In many cases, how- 

 ever, this relation must be regarded only as a rough approximation. It is neces- 

 sary to calculate the corrections to this formula and the resulting corrections to 

 the expressions of the parameters of the frequency curve. For want of time I 

 have not at present opportunity to work out these formula3 (the corrections of 

 Sheppaed are not here sufficient), but will confine myself to making an observation 

 on a single point. 



