[Extracted from the Cambridge and Dublin Mathematical Journal, Vol. vili. p. 188, 



February, 1854.] 



Solutions of Problems. 



1. If from a point in the circumference of a vertical circle two heavy particles be suc- 

 cessively projected along the curve, their initial velocities being equal and either in the same 

 or in opposite directions, the subsequent motion will be such that a straight line joining 

 the particles at any instant will touch a circle. 



Note. The particles are supposed not to interfere with each other's motion. 



THE direct analytical proof would involve the properties of elliptic integrals, 

 but it may be made to depend upon the following geometrical theorems. 



(1) If from a point in one of two circles a right line be drawn cutting 

 the other, the rectangle contained by the segments so formed is double of the 

 rectangle contained by a line drawn from the point perpendicular to the rmln-nl 

 axis of the two circles, and the line joining their centres. 



The radical axis is the line joining the points of intersection of the two 

 circles. It is always a real line, whether the points of intersection of the circles 

 be real or imaginary, and it has the geometrical property that if from any point 

 on the radical axis, straight lines be drawn cutting the circles, the rectangle con- 

 tained by the segments formed by one of the circles is equal to the rectangle 

 contained by the segments formed by the other. 



The analytical proof of these propositions is very simple, and may be resorted 

 to if a geometrical proof does not suggest itself as soon as the requisite figure 

 is constructed. 



If A, B be the centres of the circles, P the given point in the circle whose 

 centre is A, a line drawn from P cuts the first circle in p, the second in Q 



