swinging through an Arc of Finite Magnitude. 861 



It follows that the triangles LOG, MOG are equiangular, 

 since they have the angle LGO common, and the angles 

 OLG, GOM are equal. 



Hence, by Euc. vi. 4, 



LG:GO = GO:GM; 



so that LG . GM = GO 2 . 



The length of! the tangent from G onto the inner circle is 

 thus seen to be equal to GO. It follows that the system of 

 concyclic sphero-conics will project into a system of eccentric 

 circles, all having the same length of tangent from the 

 point G ; hence the system of circles has a common radical 

 axis, viz. the straight line HK. 



The introduction of the rest of the Jacobian geometry is 

 fairly obvious. 



Application to Pendulum-motion. 



We may now apply the foregoing results to investigate 

 the motion of a pendulum. 



As in the last Section, let us project the sphero-conic onto 

 a plane touching the sphere at A, one of the extremities of 

 the minor axis. Let the tangent-plane be vertical, and let 

 the straight line HK of the last Section be horizontal, lying 

 above the circle formed by the projection of the sphero-conic. 

 Let us next suppose that the circle is materialized in the 

 form of a fine wire, and let us consider the motion of a 

 frictionless bead moving on the wire under the action of 

 gravity. Suppose the bead sent off from the lowest point 

 of the circle with a velocity equal to that required to reach 

 the level of the straight line HK ; the bead will arrive at A, 

 the top of the circle, without losing all its velocity, and will 

 therefore perform a series of complete revolutions. 



From elementary Dynamics we know that the kinetic 

 energy of the bead at any point of its path varies as its 

 perpendicular distance from the straight line HK. Let us 

 call this distance/; and let </> denote the angle between B'A 

 and the line joining the point A to the bead. Then 



/=B'G-irAsiirc/>. 



