Dr. Roget on a Violation of the Law of Continuity. 205 



this precise moment it is indifferent whether we consider it 

 as existing on the one side or the other of the tangent, with 

 which the curve has now coalesced. But we may suppose 

 the same power, whatever it be, which has effected this gra- 

 dual change of curvature, to continue its operation still fur- 

 ther. A curvature in the opposite direction will now take 

 place, and the centre of this curvature will be found on the 

 opposite side to that in which it had at first been situated. 

 Its motion, in proportion as the curve continues to follow the 

 same uniform progress of change, will approach the line from 

 a distance infinitely negative. All these successive changes 

 of position in the centre of curvature are perfectly similar to 

 those of the extremity of the tangent, during the increase of 

 the arc of a circle from nothing to the entire circumference 

 of the circle ; and are, in both cases, perfectly accordant with 

 the law of geometric continuity. 



A case, however, has occurred to me, in which this appa- 

 rently universal and necessary law seems to be directly vio- 

 lated. Let the ellipse ADBE be the curve which is to be sub- 

 jected to the operation of the cause, tending, as in the former 

 instance, to diminish its curvature ; and let us trace the 

 successive changes which will , 



thereby be occasioned in the 

 position of the two foci, F, f. 

 A compressing force, for ex- 

 ample, is applied in the direc- 

 tion of the transverse diame- 

 ter, AC, BC ; or, what will 

 produce the same effect, a P-j 

 stretching force is applied in 

 the direction of the conju- 

 gate, or shortest diameter, 

 CD, CE. By the action of 

 either of these forces, the 

 form of the ellipse will be 

 brought nearer to that of a 

 circle, till it at length becomes a perfect circle, represented by 

 the dotted circumference. But it assumes this form only for 

 an instant, being merely the passage to another series of 

 ellipses, which, taking their rise from this circle, at first differ 

 but insensibly from it ; but afterwards gradually assume every 

 possible degree of compression, until they are ultimately re- 

 duced to a straight line at right angles with the transverse 

 diameter of the first set of ellipses. The former represent 

 circles that are elongated, the latter circles that are compressed 



in 



