THE POINT OF INFLEXION 189 



When tfj-0, K, - V2 x 2* -4 



a, - 2, R, - - V2 x 4^ 8 V2 = - 11-18 



#3 = 8, R 3 = - V2 x 10^ 20\/5 = - 44-72 



When x l = 0, t/j = 0, -^ = oc, tan Oj = ex, and t = 90 



x = a?j - Rj sin O x = + 4 sin 90 = 4^| 

 y = y l + Rj cos X = - 4 cos 90 = Oj 



When a? 2 = 2, t/ a = 4, -^ = 1, tan a = 1, and 2 = 45 



g-v/2 

 a; = x z - R 2 sin 2 = 2 H ^ = 10 



V 2 



e - s V 



When ff 3 = 8, t/ 3 = 8, -j- = - 



11 2 



Then tan 3 = -, sin 0, = 7=, cos 0o = 7= 



2' Vs Vs 



a; = #3 - R 3 sin 3 = 8 + ^ = 28 



V5 



3 cos0 3 = 8 j-= -32 



V5 



When aj = 0, R = 4. Co-ordinates of centre of curvature 4, 



a; = 2, R=-1M3 10,- 4 



a; = 8, R = - 44-72 28, - 32 



85. The Point of Inflexion. In the previous chapter we con- 

 sidered the case of a point of inflexion on a curve at which the 

 tangent is horizontal, but a point of inflexion can occur when 

 the tangent is not horizontal. 



Let C (Fig. 43) be such a point, and let the tangent to the 

 curve at this point be inclined to the axis of x at an angle a. 



Case I. When the angle is acute. Let tan a = m. Moving 

 along the curve from A to C, the angle is acute, and decreases to a. 



tt?y 

 Then -j- is positive, and decreases to m. 



Moving along the curve from C to B, the angle is acute, and 

 increases from a. 



Then -- is positive, and increases from m. 



