188 Dr. H. Jeffreys on 



As -j- is small when we move along a contour we can put 

 ds2 = dy, and as h 2 ds 2 = : d/jL, we find _ 



^-jhw d *- (3) 



Again, the direction cosines of the normal to the surface are 

 proportional to z± -f ~, ^-, — 1 respectively. Thus 



— -{ i+ (*' + li)V • • • • • (4) 



-- (^ + g){ i+ (^g)r- • w 



If A = A cosa, the equation of continuity now becomes 



tff 3 tan 3 a . , 9 T^ ... 



- 0/ a — = % tan- a sec a 1 -p- (b) 



= [a? tan 2 a sec a] — /i 2 tan 2 a sec a 1 x -=- I y J doc. 



" c - -' 2 • c<) 



It is fairly evident without mathematical treatment that 

 the greatest instabilities, if any, will occur for displacements 

 forming corrugations along or down the slope, and not for 

 those running obliquely. Consider first those running along 

 the slope, so that <f> does not involve y. Then A 2 = l and 



ill f O ]1" fit 



' — —r-T =ctf tan 2 asec a, (8) 



OarxQ 



where x is the horizontal distance from the top of the slope. 

 Thus where the distortion increases tan 2 a sec a it will in- 

 crease £tana, and hence the erosion. So long as the slope 

 is not very great the variation in sec a will always be small 

 compared with that in tana, and thus the surface will 

 sink fastest where the relative increase in tan ol is greatest. 

 Now considering a series of elevations of the same height 

 and horizontal extent down the slope, we see that tan a 

 is greatest on the lower side of each, and the relative 

 amount by which it exceeds the undisturbed value is 



greater the greater - — -, ~>- is. Thus if A be the top of one 



ridge, C that of the next in order downwards, B the bottom 

 of the intermediate hollow (the depth being measured 

 normally to the general slope), and J) that of the hollow 



