Mathematical Questions and Solutions, from the Educational Times. Volume 24 (Paperback)

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1876 edition. Excerpt: ...then if F, F' are centres of force, each force varying as (distance),"" prove that (1) the resultant force upon any point P on the circle is in the direction of a fixed point S on the axis OFF'; and if, moreover, the forces at the unit of distance are as (OFjJ("I) to (0F')J(""1) then (2) the resultant force is proportional to (SP)_1("-1).(PV)-i("t1), where PV is the chord through S. prove that two circles and another point may be taken arbitrarily, and that the circles abc meet the circles def in six new points which lie on the circumference of another circle. 4723. (By Prof. Crofton.)--Give a geometrical method of drawing normals to a parabola from a point on the curve. I. Solution by Professor Minchin. When three normals meet in a point, the chord joining the feet of two makes the same angle with the axis as that made by the line joining the vertex to the foot of the third. Now it is easy to show that, when the point of meeting is on the curve, the chord joining the feet of the two normals passes through a fixed point on the axis at a distance from the vertex equal to the semi-latus-rectum. Hence the construction is, --join the given point on the curve to the vertex, and from the point on the axis distant from the vertex by the length of the semi-latus-rectum draw a line making with the axis the same angle as that made by the line first drawn. We then get the feet of the two required normals. II. Solution by B. F. Davis, Nukanxa Sarkar, and others. The following property will be of use. If from any point T on the straight line x--la tangents TP, TQ be drawn to the parabola y2 = iax, the normals at P, Q, will intersect in a point O on the curve. For, adopting the -s--variable, the points on the curve the norm...