MECHANICS OF FORMA TION OF ARCUA TE MOUNTAINS 1 73 



greatest, the span of the arch must at that time be a maximum.^ 

 The point from which the arch springs upon the side away from the 

 active force thus becomes estabUshed as an abutment, so to speak, 

 and the length of the arch is maintained constant up to a Kmiting 

 stage to be presently discussed. The general shapes of simple 

 antichnes of increasing asymmetry are well known upon the basis 

 of experience, and the attempt has here been made to represent 

 successive stages arbitrarily spaced in the process of anticline 

 evolution. Six of these belong in the class denominated unsym- 

 metrical, whereas the remaining three are "overturned" (Fig. 13). 

 Throughout, the assumptions have been made: first, that failure of 

 the arch does not take place; and, second, that its length remains 

 constant — a condition which would often be reahzed in nature for 

 the first six cases, but could hardly persist long after underturning 

 had set in. 



Professor Theodore R. Running of the University of Michigan, 

 an expert in the mathematical study of curves, has at the writer's 

 request subjected this series of curves to examination. He has 

 found that to a close approximation the first seven of the series, the 

 only ones which were tested, alike possess an axial line^ which 

 bisects all horizontal chords, and that the seven may be fitted to 

 the comparatively simple general equation 





in which the obHque co-ordinate axes are the base line and the 

 bisecting axial fine, in which y° is the length of this axial line from 

 base to crown, and in which a is one-half the base. 



Competence of a relatively strong member in an anticline to lift the 

 load from inferior strata. — For the purposes of this discussion, the 

 load which rests upon an anticKne in competent strata may be 



' Cf. Willis, op. cit., pp. 251-52. 



2 Trace of the axial plane of the anticline. That this axial plane bisects horizontal 

 chords of the anticline in all stages has been often noted by the writer in studying the 

 folded schists of the Berkshire Hills of New England and elsewhere; but this character 

 had not consciously been made a basis of measurement in drawing the curves of 

 Fig. 13, which were made to accord with the shapes of anticline sections repeatedly 

 observed in the field. 



