( 312 ) 

 CHAPTER XXI. 



CONCAVITY AND CONVEXITY. 



288. THE terms 'concave' and 'convex' are commonly not 

 denned in works on the Differential Calculus, but are used 

 in their ordinary sense. The following definition however 

 has been given : " A curve is said to be concave at one of its 

 points with respect to a given straight line, when in passing 

 from that point its two branches are initially included within 

 the acute angle formed by the given straight line and the 

 tangent to the curve at that point. When, on the contrary, 

 the two branches are initially outside this angle, the curve is 

 said to be convex at this point with respect to the straight 

 line." 



289, To find a test of the convexity or concavity of a 

 curve. 



Let P be a point in a curve whose co-ordinates are x, y. 





JH JC 



Draw the tangent at P ; then if at the point P the curve be 

 convex to the axis of x, the ordinates of the curve cor- 

 responding to the abscissa xh must be greater than the 

 corresponding ordinates of the tangent at P, when h has any 

 value contained between some finite limit and zero : if the 

 curve be concave, the ordinates of the curve must be less than 

 the ordinates of the tangent. This may be deduced from the 

 definition of Art. 288 ; or if we omit that definition it must 



