290 Prof. Sylvester on Periodical Changes of Orbit, $$&. 



It may readily be found that the velocity at any point of the 

 orbit must be that due to infinity (otherwise a different and 

 much more complicated curve would result) , and then with the 

 usual notation the differential polar equation to the curve becomes 



(dOV_ c (r*-F) 2 



which is easily seen to be true of any arc of a circle. The phe- 



7/3 



nomenon to be noticed is, that when — =0, since t is the real 



dr 



independent variable, 6 does not attain a maximum or minimum, 



for it is -7- becoming infinite, not -r- becoming zero, which ac- 



dt n dt ° 



counts for -=- vanishing; accordingly — in passing through zero 



must be taken with a change of sign, which accounts for the 

 discontinuity of the orbit regarded as a geometrical curve. This 

 change of sign in the radical is very analogous to what happens 

 when we calculate the potential of a spherical shell, and trace its 

 value as the attracted point continuously receding from the 

 centre passes from within to without the shell. 



As connected with this subject of motion in a circle, I may 

 mention that Mr. Crofton has pointed out to me that my theorem 

 concerning a homogeneous circular plate whose molecules attract 

 according to the inverse fifth power of the distance, namely that 

 its resultant attraction is capable of making a particle move in 

 any circle cutting the plate orthogonally, admits of being esta- 

 blished upon my own principles without calculating, as I have 

 done, the law of the attraction (Astronomical Prolusions, Phil. 

 Mag. Jan. 1866, p. 73); for the whole plate may be shown to 

 be its own inverse in respect to any such orthogonal dividing 

 circle; i. e. the two parts into which it is divided by the plate 

 will be inverses to each other in respect to the orthogonal circle, 

 and consequently conjointly will serve to make a particle move 

 in a segment of such circle exterior to the plate*. 



purpose of explaining certain optical phenomena, a law of force according 

 to some very high inverse power of the distance transcending such limit. 

 It will be seen below that the inverse fifth power is inadmissible on this 

 ground, and is capable of leading to irreconcileable contradictions. 



* And equally it follows that a homogeneous plate whose molecules exert 

 a repulsive force following the inverse fifth power of the distance, would serve 

 to make a particle move in the interior segment of an orthogonal circle. 

 Quaere as to how the motion must be conceived to take place when the 

 attracted or repelled particle enters or quits the plate ? To fix the ideas, 

 suppose the plate attractive. The orbit described within the plate must 



