20 Mr. J. C. Glashan on the Motion of a Particle 



and all the results have still to be expressed in terms of the air- 

 thermometer. This work is in process. In the mean time, the 

 small but real and certain differences here exhibited, and which 

 have cost so much rigorous effort in their attainment, illustrate 

 the difficulties and beauties of isomerism. 



12 Pemberton Terrace, 

 St. John's Park, N. 



IV. On the Motion of a Particle from Rest towards an Attract- 

 ing Centre ; Force oc (Distance)' 2 , By J. C. Glashan, Esq.* 



THIS problem is briefly noticed by Professor Cayley in the 

 British Association Report, 1862, p. 186. Speaking of 

 rectilinear motion, he says : — " The problem thus becomes a par- 

 ticular case of that of central forces; and it is so treated in the 

 Principia, Book I. § 7 ; the method has the advantage of ex- 

 plaining the paradoxical result which presents itself in the case 

 Force oc (Dist.) -2 , and in some other cases where the force 

 becomes infinite. According to the theory, the velocity becomes 

 infinite at the centre, but the direction of the motion is there 

 abruptly reversed ; so that the body in its motion does not pass 

 through the centre, but on arriving there, forthwith returns to 

 its original position. Of course such a motion cannot occur in 

 nature, where neither a force nor a velocity ever is actually 



infinite Force oc (Dist.) -2 , = ■—, which is the case above 



alluded to. Assuming that the body falls from rest at a distance 



3 

 0? ... 



a j we have #=«(1 — eosd>), where, if n — —j=, <b is given in 



VjbL 



terms of the time by means of the equation nt = cf)— sine/). If 

 the body had initially a small transverse velocity, the motion 

 would be in a very excentric ellipse ; and the formula3 are in fact 

 the limiting form of those for elliptic motion." 



In the ' Messenger of Mathematics/ N. S. vol. iii. pp. 144- 

 149, Professor Asaph Hall sketches the history of the problem 

 and proposes a limit-derived solution. After noticing Laplace's 

 views he proceeds : — " We have now three different opinions with 

 regard to the motion of the particle : — first, Professor Cayley's 

 interpretation of Newton's investigation, which is that the par- 

 ticle reaches the point C [the centre of attraction] f, then moves 

 back toward A [the original position], and continues oscillating 

 between A and C; second, Filler's conclusion that the par- 

 ticle stops at the point C ; and, finally, the statement of Laplace, 



* Communicated by the Author. 



f The words within the brackets are ours. 



