Researches into tlie tlieocy of probability. 27 



Vj' — c = wX, 

 V,' — 2cv/ + c2 = co2(X^-f X), 

 v,' — Scvg' + 3cS/ — c^* = c.v''(X3 + _^ 



In these relations we introduce the moments about the mean, the coordinate 

 of which relating to the provisional origin is called h. The above equations now 

 assume the form 



h — c = (oX, 

 v, + (&-c)2 = co2(X^ + X), 

 V3 + 3v,(ft - c) + (6 - cf = ,.\\' + + X), 



the solution of which is 

 (20) 



Finally we have 



10 = v., : v,„ 



c = l 



^0 = [J'o '■ 



Hence we ßnd that the parameters are very easily calculated from the moments 

 of the frequency curve. 

 We now have 



(21) F{x^ ^c) = B, '\ [x] + A* (,r) + B, ^\ (,r) + . . ., 



where generally it is superfluous to know the values of i?^ and /i^. 

 Putting 



y = xio -\- c 

 we may write this equation in the form 



(21*) F{y) = B, ^ (y^^] + B, A^ + B, A^^]> + . . . 



In appljàng this formula it is necessary to define ^!^{x) by the general for- 

 mula (10), the argument being generally not an integer. Unfortunately there does 

 not yet exist a table of the function <[{x) for such values of the argument as are 

 not integer. 



As a control we derive from (20) the relation: 



(22) o)2X = a2^ 



where a signifies the standard deviation. 



For the coefficient B^ I have obtained the value 



