434 
Proceedings of the Royal Society of Edinburgh. [Sess. 
i 
iuc sin 2 /3 + yuK, where Iv is the normal cohesion between the particles. So 
long as sliding does not occur, c sin 0 cos 0 — fxc sin 2 0 must be less than 
//K ; and since c sin 0 cos 0 is greater than /me sin 2 00 the difference 
between these two quantities increases as c is increased. Sliding takes 
place as soon as this difference exceeds ,uK, and the inclination of the 
surface of sliding, therefore, is such that c sin 0 cos 0 — fie sin 2 0 is a 
maximum. Hence tan 20 = 1 j m, or 0 = 45° — 0/2. 
3. Effect of End Friction with Plates Harder than 
Material Crushed. 
Let fJL ( = tan <p') be the coefficient of friction between the crushing 
plates and the extremities of the specimen. Let c be the new compression 
FIG 2. 
yield point, and let 0' be the inclination of the new surface of sliding 
(fig- 2). 
The friction on the ends of the bar is fxc' per unit of area. Resolving 
this in the same manner as for the stress due to the direct crushing load, 
the component along the surface of sliding is fj!c sin 0 ' ; and the component 
normal to this surface is jul'c' cos 0'. Since the area of the surface of sliding 
is equal to the area of the end surface of the bar divided by sin 0', the effect 
