ACTING FACETS OF ARTICULAR SURFACES. 255 



its length. If, now, the generating spiral increases its dimen- 

 sions, so that each linear increment corresponding to a given 

 angular increment shall vary as the existing dimensions of 

 the spiral itself, then the longitudinal curvatures of the sur- 

 faces developed that is, the lines of curvature resulting from 

 the revolution of the continually-increasing generating spiral 

 round, and the gliding of it along, the fixed axis, must also be 

 equiangular spirals, with a constant ratio determined by that 

 of the generating spiral. 



26. The opposite surfaces, thus simultaneously generated, 

 would evidently be congruent screwed surfaces, representing 

 respectively the concave and convex elements of a conical 

 screw-combination. The curvature in the direction of the 

 thread of such a screw-combination would possess the cha- 

 racter of the equiangular spiral ; while the curvature across 

 the thread would possess corresponding characters. 



27. It is evident that the concave and convex elements of 

 a screw combination of this kind would be fully congruent 

 when in apposition, and that their axes would be coincident. 

 It is also clear that any attempt to unscrew a combination 

 of this kind, while the axes of its two elements are retained 

 in a right line, would at once render its entire opposite sur- 

 faces incongruent, in fact separate them from one another. 

 If, on the other hand, it were possible to diverge the axes of 

 the two elements from one another at the same time that the 

 elements are unscrewed, then there would result from these 

 combined movements a gliding of the opposite surfaces upon 

 one another over a succession of extents, which would 

 diminish in area in terms of the constant ratio of the generat- 

 ing spiral, until finally a minimum of contact of congruence 

 would obtain that is, to use the terms already employed, 

 the screw- combination would be in its negative phase, and its 

 elements in their negative positions. On reversing the com- 

 bined movements that is, on screwing the combination into 



