Prof. Sylvester on Periodical Changes of Orbit, fyc. 289 

 Thus, ex. gr., if the angular sector be 72°, the orbit will be 



Pi Q 2 P 2 Q 3 P* Q 4 P 4 Q 5 p 5 Qi ? 2 Q 2 ^3 Q 3 P 4 Q 4 P 5 Q5 P. &c. 



In like manner, if the angle included between the tangents be 



any commensurable part of 360°, as — 360°, where m and n are 



integers prime to one another, the orbit will be a closed one 

 containing mn alternate segments, or mn entire circles, according 

 as n is even or odd. By taking n even and giving m any arbi- 

 trary odd value, a waving line will be produced forming an ori- 

 ginal and, I think, elegant pattern for a circular lace border. 



m 

 For this purpose — should not be too small, in order that the 



disproportion between the alternate circular segments and the 

 ratio of the border to the interior may not be so great as to 

 offend the eye ; and m not too great, in order that the traces of 

 the pattern may not become too complicated. 



I conjecture that [?n = 3; n=64] and [m = 5; n = 128], pro- 

 ducing respectively 3 and 5 twists, and 5 or 6 and 6 or 7 flex- 

 ures within a quadrant of each twist, would be eligible systems 

 for the purpose. In general n ought to be even, and m a large 

 moderate odd integer. 



If the angle between the tangents to the circle from the force- 

 centre be not an aliquot or commensurable part of 360°, the 

 orbit will be a non-reentrant curve intersecting itself an infinite 

 number of times. Similar or analogous conclusions are of course 

 applicable to every case where the orbit, seemingly indicated by 

 the equations of motion, is an oval, or, more generally, any curve 

 to which tangents admit of being drawn from the force-centre, — 

 a self-evident (now that it is stated) but none the less a very sur- 

 prising feature in the mathematical theory of central forces. I 

 say mathematical ; for it ought in fairness to be observed that 

 since it is impossible to conceive a force of infinite magnitude 

 resulting from the attraction of a finite mass, the question in- 

 volves not so much a discussion of any real phenomenon, as of 

 the principles of interpretation applicable to an extreme case, 

 valueless as to the establishment of a distinct independent con- 

 clusion, although not without latent importance as a safeguard 

 against errors which might flow from the adoption of an erro- 

 neous mode of interpretation*. 



* If we accept the very reasonable axiom that no law of force is admis- 

 sible which would involve the consequence of a finite mass exerting an 

 infinite attraction at a finite distance, we can find an a priori limit to the 

 negative exponent of the power of the distance which can possibly express 

 any law of force in nature. If my memory serves me aright, a distinguished 

 rising French analyst, in contravention of this axiom, has assumed, for the 



