ACTING FACETS OF ARTICULAR SURFACES. 257 



linearly equal to every lesser development of it. And as all 

 these geometrically-similar and linearly-equal portions of 

 curvature are, 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 passage from 

 the positive, to the negative phase, each greater transverse curva- 

 ture, when it advances on the next opposite lesser curvature, 

 cannot coincide with it as a whole, but "by the transverse gliding, 

 as much of its polar extremity as is geometrically similar and 

 linearly equal to that next opposite lesser curvature will coincide 

 with it, while the remaining portion of its antipolar extremity pro- 

 jects uncovered leyond 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 terms of the common ratio of the generating curve, 

 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 

 similar, 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 progressively diminishing, or increasing, portions 

 of similar and equal transverse curvature, involves a corre- 

 sponding series of progressively diminishing or increasing 

 similar and equal portions of longitudinal curvature, and 

 consequently a resultant series of progressively diminishing, 

 or increasing, similar and equal areas of curved surface for 

 contact on the opposite elements of the articular coupla 



31. It is also evident that we may assume the generating 



s 



