20$ EOOK OF MATHEMATICAL PROBLEMS. 



to the curve is - , where r is the distance from the fixed 



r p sin <f> 



point to the point of contact, <f> the angle between this distance 

 and the straight line, and p the radius of curvature at the point 

 of contact. 



997. If the curve 



- = 1 4- sec a sin (0 sin a) 

 roll on a straight line, the locus of the pole is a circle. 



998. If (a, /?, y) be areal co-ordinates of any point on a 

 curve, p the radius of curvature at that point, 



d'afdf! dy\ d^(3 (dy _ da\ <*y (da __dp\\ 

 7F \dt dt) + fit* \dt dt ) dt* \dt dt )} 



a, 6, c being the sides, and k the area, of the triangle of refer- 

 ence. 



999. If tvro tangents to an ellipse be drawn intersecting 

 a given length on a fixed straight line, the locus of their inter- 

 section is a curve of the fourth degree having contact of the 

 third order with the ellipse at the points where the tangents 

 are parallel to the given line : trace the curve for positions of 

 the fixed straight line in which it intersects the ellipse (1) in 

 real points, (2) in impossible points, the length intercepted being 

 equal to the parallel diameter. 



1000. Find the envelopes of 



(1) x cos 8 6 + y sin 8 = a> 



being the parameter in each case. 



