76 SOLUTIONS OF PROBLEMS. 



The circle A, the radical axis, and one position of the line Pp, are given 

 by the circumstances of projection of P and p. From these data it is easy to 

 determine the circle B by a geometrical construction. 



It is evident that the character of the motion will determine the position 

 of the circle B. If the motion is oscillatory, B will intersect A. If P and p 

 make complete revolutions in the same direction, B will lie entirely within A, 

 but if they move in opposite directions, B will lie entirely above the radical axis. 



If any number of such particles be projected from the same point at equal 

 intervals of time with the same direction and velocity, the lines joining successive 

 particles at any instant will be tangents to the same circle ; and if the time 

 of a complete revolution, or oscillation, contain n of these intervals, then these 

 lines will form a polygon of n sides, and as this is true at any instant, any 

 number of such polygons may be formed. 



Hence, the following geometrical theorem is true : 



" If two circles be such that n lines can be drawn touching one of them 

 and having their successive intersections, including that of the last and first, 

 on the circumference of the other, the construction of such a system of lines 

 will be possible, at whatever point of the first circle we draw the first tangent." 



2. A transparent medium is such that the path of a ray of light within it is a given 

 circle, the index of refraction being a function of the distance from a given point in the 

 plane of the circle. 



Find the form of this function and shew that for light of the same refrangibility 



(1) The path of every ray within the medium is a circle. 



(2) All the rays proceeding from any point in the medium will meet accurately in 

 another point. 



(3) If rays diverge from a point without the medium and enter it through a spherical 

 surface having that point for its centre, they will be made to converge accurately to a point 

 within the medium. 



LEMMA I. Let a transparent medium be so constituted, that the refractive 

 index is the same at the same distance from a fixed point, then the path of 

 any ray of light within the medium will be in one plane, and the perpen- 



