288 Prof. Sylvester on Periodical Changes of Orbit, fyc. 



January Number of this Magazine, where I alluded, in passing, to 

 the case of a body acted on by a central force capable of making 

 it move in a circle exterior to the force-centre, and fell into the 

 not unnatural error, which has since been pointed out to me, and 

 which is obvious on a moment's reflection, of stating that on 

 arriving at a point where the motion points to the force-centre, 

 i. e. at the point where the tangent to the circle passes through 

 this centre, the particle would go off in a straight line on account 

 of the motion and the force coinciding in direction. But it is clear 



that since the instantaneous area \p* -=- remains finite at such point, 



it cannot abruptly become zero ; the radial velocity becoming 

 infinite, does not entitle us to reject the transverse part which 

 remains finite; thus the radius vector p will continue to revolve 

 in the same direction as before it reached the tangential point ; 

 it will therefore swing off to another curve, so that the true orbit 

 will possess an inflexion at that point. The new curve, it may 

 easily be proved, will be a circle equal to the former, and related 

 to it in the. manner following : let us suppose to be the force- 

 centre and two tangents drawn from O to meet the original circle 

 in A and B, so that the line A B divides the circle into two un- 

 equal segments, and that the particle has been travelling, say in 

 the upper segment, from A to B ; draw the angle B O C equal to 

 the angle A OB, and in it place a circle equal to the former, 

 touching the part O B, C in B and C ; then the particle will 

 describe the lower segment of this new circle ; and so in like 

 manner, after reaching C, will undergo a new inflexion at that 

 point and pass on to a new circle touching C and D, the 

 latter inclined to the former at the same angle as C to OB 

 and B to A. Thus, if we repeat the angular sector A B 

 indefinitely, and in each such sector place equal circles touching 

 the rays of the sector, and call their upper and lower segments 

 P, Q respectively, the particle will describe the successive arcs 

 P u Q 2 , P 2 , Q 3 , P 3 , . . . . ad infinitum. If the sectorial angle be 

 an even aliquot part of 360°, the complete orbit will be a single 

 anautotomic broken curve returning into itself, as, for instance, 

 if the sector be 90° the orbit will be l\, Q 2 , P 2 , Q 2 , P 3 , Q 3 , P 4 , 

 Q 4 , Pj, Q^, .... If the angle be an odd aliquot part of the same, 

 the orbit will be a line returning into but crossing itself as many 

 times as there are circles, so that in fact the whole of each circle 

 will be described in a complete period, viz. the upper and 

 lower segments alternately in the first half period, and the lower 

 and upper in the second half thereof, the period being double 

 the time of a revolution if the latter is defined as the interval 

 between the body leaving and returning to any the same point. 



