Researches into the thcoiy of probabiHty. 7 

 Choosing the origin of the ^.'-coordinates arhitrarily, we put 



(2) jj.; = fa;' /''(,r)7te. 



00 



On the other side we put 



+ '-^ 



QO 



so that 



[J.. = IK - (;) h + (^) aL, - (^) + . . . 



where (;), (2), (s) designate the binomial coefficients. 



If tiie quantity h is Ivnown, we know also the values of [x^, [j,^, [Xj,, . . 

 Now h is given by the equation 



3) [}.lh = ]i.[. 



We then have 

 (3*) 1^,^^ = ^-.. 



and the quantities yl.j, A,^. . . . have the values 



Aq = [Xy , 



(4) ]4^,= [x,- 3a^.x,, 



jô Ar-, — — [J.. + 10 a- [X.,, 



|6 = [x,. — 15 a-' [x^ + 1 Ô ci« [x„ , 



The quantities (x^, jx^, [Xg, . . . are named the moments, taken in respect to 

 (or about) the point 6, of the curve y = F{.r) of the first, second, third, . . . 

 order. When these quantities are calculated, it is easy to calculate the values of the 

 coefficients A^^, A.^, A^, . . . according to the formulte (4). 



As to a it is named by English writers on probability the standard deviation. 

 German mathematicians generallv call it mean deviation or mean error. As to tp {x), 

 it is the form of the prohahiJity function generally used by Pearson. I find that 

 this form is to be preferred before the usual Gaussian form 



, , Iv — li' ix—bf 



|/ TU 



where Ä- is called the measure of precision. The difference is naturally only a formal 

 one, but being a length (supposing x to be considered as a length), is easier to 

 conceive than the quantity h. I will in this connection remark that the so-called 

 probable error may without regret be removed from the practical applications of 



