Asymmetrical Probability -Curve. 91 



measured from an origin such that the average value of each 



of ihe q's, and therefore also the average value of Q, vanishes 



(Q being omitted) ; y Ax is the proportional number of 



c 2 

 values of Q occurring between x and x + Ax ; -»- is the sum 



of the n quantities each of which is the mean square of error 

 (measured from the average value or centre of gravity) for 

 one of the elements Q^. 



The asymmetrical probability- curve is a second approxima- 

 tion which may be written 





where #, y, and c have each the same meaning as before ; j is 

 the sum of the n quantities each of which is the mean cube 

 of error for one of the elements Q'g. I propose to give a 

 new proof of this formula ; after first adverting to one or two 

 old proofs. 



I. The formula may be derived from an analysis which 

 Todhunter, following Poisson, has indicated. Todhunter 

 inquires what is the probability, P, of the value of a certain 

 quantity E occurring between c + rj and c—rj; E being = 

 7i e i + 72 e 2 + • • • 5 where ry lf <y 2 > • • • are constants, and each of 

 the quantities e l3 e 2 , &c. fluctuates between given limits, a and 

 b, according to a law of frequency which is of the form 



y=f.(x) ; the value of I xf.(x)dx being k.; the correspond- 



ing value of mean second powers being k/, of mean third 

 powers hi f . For a first approximation to P, Todhunter finds 



P = — ^7= I e 4*» dv ; 



1 C* -Jt 



= r^ e 



where 1 = ^.^, and 2/c 2 = X r yi 2 {h! —h?) . This coincides with 

 the first approximation given above* when ki, k 2 each = 0, 

 and when the interval 2rj is indefinitely small. For then P 

 reduces to 



1 z± 



7=04*2 x 2t7 ; 



2/C^TT 



where c 2 may be replaced by our x 2 , 2tj by our Ax, and 

 4« 2 = our c 2 . 



For a second approximation Todhunter indicates as the 



* P. 90. 



