curve may therefore be regarded as the limiting case in 



the particular direction. 



6. Critical points of curves.— By successive differentia- 

 tion, equations may be formed giving the critical points of 



the complete curve y = x m e n * v . 

 For brevity, writing E for e nxP and P for *p& , the 



successive derivatives, as far as the third inclusive, are: — 



f(x) =ri (lib) 



f'(x)= X"- l E{m+P] (12) 



n*)= x m - 2 E[m(m-l) + P{2m + p-l}+P 2 ~\ (13) 



f'(x) = x*-*E [m(m - l)(m - 2) + P{ 3m(m - 1) + 3m(p - 1) 



+ (p-l)(p-2)}+P 2 {3m + 3(p-l)]+P 3 l...(14) 

 Thus from (12) the maximum or minimum value of y will 



be given by putting f (z) = 0, that is,— 



m + nps* = 0, or z m = (-m/*tp)J (15) 



which when p = l, becomes 



«m = -W« (15a) 



£ m denoting the value of the abscissa for a maximum or 



minimum value of y. 



When x has this particular value viz, that in (15), equation 



(13) reduces to 



d* V /dx* = -mpr- 2 c wP (16) 



which, when m and p are of the same sign, is negative ; 



and when they are of opposite signs, positive. Hence from 



the criterion of convexity and concavity, for a maximum 



value, m and p must have the same sign, and for a minimum 



value opposite signs. 

 The points of inflexion occur when d*yfdx* = 0, which 



we see from (13) occurs when 



P 2 + P(2m + p-1) + m(m-l) = (17) 



The roots of this equation are 



P=np.r p = i{-(2m+p-l)±4(2m+p-l) 2 -4m(w-l)]}...(l8) 



