f x v-^ e ^vdx = npe UT » or /ar m c ux " T dx = n(w + l)e ,lx (72) 

 as can be seen by differentiating the right-hand member, 

 see (8) herein, or from (68) since m—p + 1 is then zero. 

 When m = 1 - n or p = 1 - m we have from (61) 



/*' 



1x1 </.'• -~ 



and from (63) the same integral is equal to 

 2— p 1 2 + 2(2+p) '"' 



+ (-DMnpq ?)' +etc (74) 



2(2 + p)...)2 + (r-l)Pj 

 When i» = - 1, the term e nxP may be expanded, the 

 integration term by term then giving 



je^ d *= logx + » * + •! £-+... + £ **" + etc(75) 

 J x s 1! p 2! 2p r! rp 



From (68) we also obtain 



/** * - •* i l + JL + _2 ! + _ + _rl + etc I (75a) 



x npx* \ uxp (**» ) 2 («*»? I 



In general, if the relation between m and p can be simply 



expressed, some simplification of the formulas will be found 



to facilitate computation. 



12. Particular values of abscissa.— Several values of the 

 abscissa besides those already referred to (viz: maximum, 

 minimum, or mode, points of inflexion, points of most rapid 

 change in direction of tangent, etc.) are required. The 

 most important of these is the abscissa, x c say, of the 'cen- 

 troid vertical " i.e. of the me,,,, of the distribution. This 



1 Since the function proposed for the determination of skewness 

 i* (\,-\m) :/|Y(.\)\VyJ /f(\)d X \* the ordinates X being 

 measured from the intersection of the centroid vertical with the 

 axis. (See W. Palin Elderton's 'Frequency Curves and Correla- 

 tion,' pp. 10, 11). 



