DIFFERENTIAL CALCULUS. 205 



980. In the curve r = a sec*0, prove that, at a point of 

 inflexion, the radius vector makes equal angles with the prime 

 radius and the tangent; and that the distance of the point of 

 inflexion from the pole increases from a to a ^/c, as n increases 

 from to oo . If n be negative, there is no real point of 

 inflexion. 



981. SY is the perpendicular from the pole S on the tangent 

 to a curve at P\ prove that when there is a cusp at P, the circle 

 of curvature at Y to the locus of Y will pass through S: also, 

 that when thore is a point of inflexion at Y in the locus of }*, the 

 chord of curvature at P through S will be equal to SP. 



982. The general equation of a curve of the fourth degree 

 having cusps at A, B, C is 



&*&& 



A BC being the triangle of reference. 



983. The equation 



* _* 

 y = C* -f 6c * 



will represent a catenary if 46c = a f . 



984. If x, y be rectangular co-ordinates of any point on 

 \e, p the radius of cur\; liat point, <f> the angle 



li the tangent at the point makes with a fixed straight line, 



985. The centre of curvature at a point /' of a parabola ia 0, 



iit angles to 01' m* ting the focal distance of 



/' in ',' ; a the radius of curvature of the evolute at is 



tO ZOQ. 



