264 On Torsion-Structure in the Alps. 



Appendix. 

 (a) The curves shown in fig. 2 have some interesting 

 properties. Those figured are drawn so that the angle be- 

 tween two consecutive radii is 10° ( =^1. 



It is easy to see that the equation to the system can be 

 written 



y = xtan^(x — c) (1) 



where c is a constant for each curve, but varies for different 

 curves : its value is marked on each curve shown. 



(6) If we assume that the curves are normal at each point 

 to the resultant force at that point, then if X and Y represent 

 the component forces, we can write 



dy X , 9 , 



From (1) we have 



g = tanf g (*-c)+*"x * .(l + tan 2 ^(*-*)) 



2 + ? ,2 

 X 



_ y_ , ir tf+y 



* 18 



r 2+ 18 



x 



where y 2 = a? 2 + ?/ 2 . 



Thus, by means of (2), we may put 



X=-K|-K.f 8 , 



where K is an arbitrary constant. 



The displacing forces can therefore be considered as 



77" 



(a) A constant force = — K^ acting parallel to the axis 

 of*. lb 



(/3) A force = — perpendicular to the radius vector. It 



will be noted that X and Y fulfil the " equation of continuity " 

 for the case of forces in a plane, namely, 



dX dY =Q 



dx dy 

 (c) Further, the. curves represented by (1) have their 



