ACTING FACETS OF ARTICUI.AK SURFACKS. 2ri7 



linearly eijual to eveiy lesser development of it. And as all 

 these geometrically-similar and linearly-equal portions of 

 curvature arc, from the assumed relations of the generating 

 spiral, necessarily collocated in a regular series along the 

 two elements of the couple, it follows that in the passarje fro77i 

 the 2->ositivc to tlie tufjatice phase, each greater transverse curva- 

 ture, when it advances on the ncj:t opposite lesser curvatiire, 

 cannot coincide with it as a whole, hut hy the transverse gliding, 

 as much of its 2^olar extremity o-s is geometrically similar ami 

 linearly equal to that next opposite lesser curvature will coincide 

 loith it, xvhile tlie remaining portion of its antipolar extremity p7'o- 

 jeets uncovered beyond the antipolar extremity of the lesser. The 

 same relations subsisting between all the other greater trans- 

 verse curvatures of the one element, and those next lesser 

 transverse curvatures on the opposite element, all the remain- 

 ing transverse portions of linear coincidences go on dimi- 

 nishing, in tenns of the common ratio of the generating cur\'e, 

 until the negative phase is reached, or loss of contact occurs. 

 On reversing the combined movements during the passage 

 from the negative to the positive phase, portions geometrically 

 simOar, and linearly equal transverse curvatures, increasing 

 in terms of the common ratio, successively coincide ; while the 

 proximal and distal extremities of the twin-elements respec- 

 tively approach one another, and their distal margins approxi- 

 mate till complete congruence is again attained. 



30. It is evident that the principle, which provides for a 

 succession of progi-essively diminishing, or increasing, portions 

 of similar and equal transvei-se curvature, involves a con-e- 

 sponding series of progressively diminishing or inci-easing 

 similar and equal portions of longitudinal curvature, and 

 consequently a re.sidtant series of progressively diminishing, 

 or increasing, similar and equal areas of cu^^'ed surface for 

 contact on the opposite elements of the articular coupla 



'M. It is also evndent that we may assume the generating 



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