256 CURVATURES AND MOVEMENTS OF THE 



its stable or positive phase the two elements would glide on 

 one another over successive extents of congruence or contact, 

 which would increase in dimensions in terms of the constant 

 ratio of the generative spiral, until both surfaces become con- 

 gruent throughout i.e., until the elements are again in their 

 positive positions. 



28. The combined movements i.e. 9 the movements along 

 the thread, and the inclination and consequent divergence of 

 the axis of the elements, or, what is equivalent, the move- 

 ment across the thread are only possible under the condition 

 that the structure of the combination shall consist of rigid 

 materials, along a limited extent only of a single spire or 

 whorl, so that the movements on that side of the whorl shall 

 not be counteracted by those on the opposite side. It is this 

 condition of applicability which determines the comparatively 

 small extent of whorl in the greater number of articular 

 couples, and more especially in their restricted elements. 



29. The combined movements being thus provided for, 

 the successive longitudinal and transverse adaptations must 

 occur in the following order : The elements being in their 

 positive phase, it is evident that during the passage to the 

 negative phase successive transverse curvatures in the one 

 element must pass off and become unscrewed at the proximal 

 extremity of the couple ; while at the same time successive 

 transverse curvatures of the opposite element must be left 

 uncovered at the distal extremity of the couple. With 

 regard to the mode of adaptation of those successive portions 

 of the elements still in contact or covered by one another, it 

 is to be borne in mind that the successive transverse curva- 

 tures are successive developments of a given extent of a given 

 equiangular spiral ; they are all, therefore, geometrically dis- 

 similar, as well as linearly unequal ; but from the law of the 

 equiangular spiral, every greater development of a given 

 extent of curve contains a portion geometrically similar and 



