314 POINTS OF INFLEXION. 



ordinate of this tangent corresponding to the abscissa x-}- h. 

 Then 



y v = () + ftf(a>) + |V' (a + Oh], 



therefore y\~Vz 77 0" 



M 



Hence, if <f>" (a?) be not zero, the sign of y y a will, if 

 /z, be small enough, be the same as that of </>' (a?), whether 

 h be positive or negative, and the curve cannot cut its 

 tangent. Therefore if there be a point of inflexion, we must 

 have <f>" (x) = 0. Suppose this condition satisfied, then 



and this expression changes its sign when h does, provided 

 </>'"(#) be not zero. If <j>"(x) be zero, it may be shewn that 

 <""(#) must also vanish ; and generally if for a certain value 

 of x several of the successive differential coefficients of y 

 vanish, beginning with the second, there is a point of in- 

 flexion if the first differential coefficient that does not vanish 

 is of an odd order. 



Since generally at a point of inflexion ~ z vanishes while 

 y- 3 is finite, -7^ changes its sign. For ~^ is the diffe- 



rential coefficient of -v ; therefore, by Art. 89, if -Tp be 



cPy . ... , . d s y ,. <ffy 



positive -T3 increases with x, and it -& be negative -~ 



d z v 



decreases as x increases. Hence -r4 must pass from negative 



dx 



d?v 

 to positive if -=4 be positive, and from positive to negative if 



d 3 y 



-7^ be negative. 



