6i 



Trof. Milne — Acrosa Europe and Asia. 



3rd. The power that material when in motion has to roll to 



lower levels, and to disengage other matter that it may meet 



with in its descent. 



Here we see three of the parts into which the subject divides 



itself, and each of these has to be considered relatively to slopes of 



varying inclination. 



Part I. — The best way to look at Part I. is to take the simple cases of bodies 

 l)ing- upon inclined planes. The force tending to pull the body down such a plane 

 is its weight ( IV), multiplied by the sine of the angle (a) of the inclination of the 

 plane, or TFsin a ; but as the inclination of the plane increases, sin a also increases : 

 therefore fV sin a increases. That is to say, if we have two bodies of equal weights 

 on planes differently inclined, the force tending to pull the body down the steeper 

 plane is greater than the force tending to pull the body down the plane which is less 

 inclined. 



The reason of thus stating a self-evident condition is for the purpose of comparing 

 the relation existing between these forces on two different planes, which we see must 

 be proportional to sin a. Thus if we take planes with slopes of 1 0°, 20°, 30°, 45°, 60°, 

 and on each of these there rests a stone of weight W, the force tending to pull this 

 stone down will be, 



TFsin 10°, TTsin 20°, TTsin 30°, JFsin 45°, and TFsin 60°, 

 or 7F-173, TF -342, TF-500, TF-707, TF-866. 



From these few cases we see that for low slopes, say up to 30°, the force tending 

 to pull the stone downwards increases approximately proportionately to the angle ; 

 that is, for double the inclination you get double the force ; but on slopes of a steep 

 inclination, as compared with those of a low inclination, this rate of increase is not 

 so rapid ; thus the force tending to pull a stone down on a plane of 60° is not six 

 times the force tending to pull a stone down on a plane of 10°. 



This result will be much more strikingly illustrated if we take into account the 

 initial resistance to motion, where the force at starting will be TF sin a — jix W cos a, 

 Ij. representing the initial resistance to motion, whether it is of the nature of friction 

 or the resistance offered to the breaking off of rocky masses. 



Pakt II. — Before considering the effect that matter has when in motion down an 



incline, we must recognize the fact that a stone when it obtains motion upon a steep 



mountain, as in falling from a rock, the rolling over of a stone usually would obtain 



more Kinetic energy at its start, than one upon a moimtain of less inclination would. 



Thus upon an inclined plane AB, a particle falling from the summit of a rock a c 



to the point b upon the plane AB, 

 a would usually have a much greater 



^ height to fall upon a plane of steep 



inclination, than it would upon one 

 where the inclination was more mode- 

 rate, and therefore at its starting to 

 roU towards B, would also have more 



energy. 



Calling ah=h; ac=-K\ the in- 

 clination of the plane =a; then /* = 

 JTsec a, from which we see the rocks 

 upon two different planes being 

 similar, or ac being constant, then 

 this height varies as the secant of the angle of the inclination of the plane. There- 

 fore a rock thus starting in a manner analogous to that considered, either by tumbling 

 forward or actually falling, has at the commencement of its course more energy to 

 do work as a stone in motion — this energy varying as the secant of the angle of in- 

 clination of the plane. 



Pakt III. — The power that a stone has in rolling to lower levels and overcoming 

 obstacles in its passage over the sides of two mountains which differ only in inclina- 

 tion, might be compared by the distance they relatively travel. It is evident one on 

 a steep slope would tend to roll farther than one on a gentle slope, and the vertical 

 distances down which these blocks would travel before coming to rest would measure 

 • the relative rates of this sort of degradation,— because the mountain which rolled 

 its material farthest down would tend to become level the quickest. 



