66 C. Davison — Form of Rock- Surface tinder a Talus. 



Let A B, the original face of the cliff, be inclined at an angle 

 a to the horizon. In any given interval of time, suppose the layer 

 A .By - ! a x to be worn from the face of the cliff, and the material 

 spread at the foot of the cliff as a x b x c x , the surface b x c x being plane 

 and inclined to the horizon at an angle (3. Let AB=h, Aa x =x x , 



Now, the parallelogram ABf x a x is equal to a rectangle whose 

 base is Aa x and height h sin «, 



.'. area A Bf x a x =1ix x . sin a, 

 and triangle a x b x c x =^ a x c x , a x b x . sin a, 

 sin (a—/3~) 

 sin 8. 

 , sin (« — 8) sin a. 



= 2 Vl • . 



2 * sin 8 



. 2 ft sin 8 



.. y x =— — . a3i. 



sin (a — 8) 



During a second interval, equal to the first, let another layer of 

 the same thickness be worn from the face of the cliff b x f x , and be 

 deposited over the debris at the base, and inclined at the same angle 

 8 to the horizon, so as to raise the surface of the talus to b 2 c 2 . Let 

 a 2 b. 2 —y 2 , and Aa 2 =x 2 = 2x x . 



Now, area f x b x b 2 f 2 = (h — y x ) x x . sin a, 

 and area c x c 2 b 2 b x = a 2 b 2 c 2 — a x b x c x — a x b x b 2 o 2 , 



= ^ a x u x . . — — — . a x b x . sin a, 



2 sin (a — 8) sin a ^ 2 sin (a — 8) sin a 

 sin 8 sm 8 



sm /3 sm /3 



sin /3 ^ -, sin /3 



y*=4:7i.- ^—.x x =2h, 



sin (a— 8) sin (a— 8) 



Proceeding in this way, we see that after n layers have been 

 worn away, we have 



2 o t sm y 3 



sin (« — /3) 



Now, let each interval of time be supposed very small ; then the 

 layer disintegrated during each interval is very thin, and the form 

 of the rock under the talus is a curve which has the property 



PN*=2h. s[n(B - .AN, 

 sm (a— 8) 



and is therefore part of a parabola whose axis is horizontal. The 

 initial face of the cliff is a tangent to the curve at A, and the final 

 surface of the talus, when it reaches the top of the cliff at Q, is 

 a tangent to the curve at that point. 



