(-2P +1 m)r> - ;i-2m-p- 44mp+(p-l) 2 ] (• p 



+ {l-2m-p+v[4mp+(p-l) > ]}|...(20) 



from which, wlien p is known, in can be so determined as 

 to give the condition of equidistance of the points of in- 

 flexion from the maximum or minimum ordinate. This 

 condition of equidistance is in all cases, independent of the 



When m = l, equation (19), for the point of inflexion, gives 



and since when m has this value, that of the maximum 

 ordinate is 



*=-<l/*f>)I, 

 there is then no point of inflexion between the origin and 



With the ordinary algebraic conventions, if imp be 

 negative and numerically greater than (p-l)% the points 

 of inflexion are imaginary; and again if m and np have 

 unlike signs the value of x for which f{x) is a maximum or 

 minimum, viz, ( - m/np) v , is in all cases real ; but if m and 

 np have like signs the value may be imaginary. 



There are other points in the curve which may be im- 

 portant statistically, viz, those at which the tangent is 

 changing its direction most rapidly. The points of inflexion 

 are those at which the rates of increase of f(x) are either 

 algebraically greatest or least, and the maximum and 

 minimum values of f{x) are those at which the rates of 

 increase are stationary. Hence the graph of f(x) is a curve 

 crossing the axis of abscissae at the values of x which give 

 a maximum or a minimum, and has, for its abscissae of its 

 own maximum and minimum, those values of x which give 

 tiie |n»i:ii - .; inflexion of the original curve. The abscissa; 

 of the points of inflexion of the graph of f\x) are those of x 



