from Rest towards an Attracting Centre. £1 



that the particle passes through the point C and oscillates be- 

 tween A and B" [a point equidistant with A from C]. 



On page 15.2 of the same volume of the ( Mesengcr } Professor 

 Cayley writes : — " I quite admit that, considering (with Professor 

 Hall) the attracting particle as split into two equal particles 

 placed at equal distances above and below the centre C, the mo- 

 tion when the distances become infinitesimal is a motion not as 

 above, but backwards and forwards along the entire line AB; but 

 it remains to be seen whether at the limit this can be brought out 



ax ll 

 as an analytical solution of the differential equation -^ = ^. 



Possibly this may be done; and I remark as an objection, not 

 to the foregoing as an admissible solution of the problem, but to 

 its generality as the only solution, that in writing x = a{\ — cos <£), 

 and assuming that (/> is real, I in effect assume that x is always 

 positive. But the burthen of the proof is with Professor Hall, 

 to show that there is an analytical solution in which x acquires 

 negative values." 



A problem that has received three inconsistent solutions from 

 such mathematicians as Euler, Laplace, and Cayley is certainly 

 worthy of notice for this alone ; but this one is besides important 

 in the theory of plane algebra, and in the theory of limits invol- 

 ving discontinuity. 



To Professor Cayley's solution and to the last of the above 

 quotations I object : — 



(1) The motion from absolute rest is not the limit of motion in a 

 fixed ellipse {nor in a fixed parabola ivith vanishing latus rectum). 



d^K Hi 



(2) The equation tt^ = \ does n °t represent the limit of 



motion in an ellipse ; neither does it represent the motion under the 

 conditions proposed in the problem. 



(3) The required equation is -^-5 = — — ~, 7=§r« 



df x( + vx 2 ) 



(4) The problem may be considered as the limiting case of" a par- 

 ticle moved from rest under the action of a spherical surface all the 

 parts of which attract the particle ivith forces varying inversely as 

 the square of the distance, find the law of the motion of the particle"*. 



In support of (1) it is sufficient to note that in elliptic mo- 

 tion, even at the limit [elliptic limit), both tensor and versor 

 vary continuously and dependently, while in the case of motion 

 from rest the versor is discontinuous, remaining absolutely con- 

 stant for all variations of the vector that does not carry it through 



* But (?) ; for in this case the force at a point within the spherical sur- 

 face is zero ; whereas in the original problem the force continually increases 

 and becomes infinite at the centre. — Ed. 



