ME. HOPKINS ON THE THEOEY OF THE MOTION OF GLACIEES. 
693 
In exactly the same way we find, by taking the moments with reference to the axes 
parallel respectively to those of y and 
B"=C". 
By means of these three relations the six accented quantities are reduced to three inde- 
pendent quantities. 
15. “ Let us now conceive a plane to meet the three coordinate planes so as to form 
with them a tetrahedron, whose vertex is at the origin P. Suppose the exterior normals 
to the three faces formed by the coordinate planes to point respectively towards the 
positive directions of x, y, and z ; and let a, j3, y be the angles which the normal to 
the base of the tetrahedron makes with the coordinate axes of x, y, z. Also, let s denote 
the area of the base, and s’, s", s'” the areas of the sides of the tetrahedron perpendicular 
respectively to the axes of x, y, z, all these quantities being indefinitely small. 
“ Again, let^s denote the whole resultant force on s, and let X, (jb, v be the angles 
which its direction makes with lines parallel to the axes x, y, z, this direction being 
exterior to the tetrahedron. Then in order that the tetrahedron may be in equilibrium, 
we must have 
yjs . cos A = As' -j- his” -f hi ’s'”, 
y)s. coSjW/=Bs" +B's' 
but 
2 )s . cos V =C s'” -j- Os' -p C”s” ; 
r 
s 
-= cos a, 
s" s'" 
--=cos(3, -=cosy; 
making these substitutions, and also putting 
B"=C"=D, 
A"=C' =E, 
A'=B'=F, 
we shall have 
y). cos X =:A cos a+F cos |8+E cos y,' 
p . cos B cos i3-hF cos a +D cos y, 1 
p. cos {' = C cos y-f-E cos afi-D cos j8. 
. {a.), 
formulse in which the notation agrees with that of M. Cauchy*. 
16. “ If }> denote the angle between the direction oi p and the normal to s, we shall 
have p . cos ^ for the whole normal force acting on the area s in a direction exterior to 
the tetrahedron, and p . sin § the whole tangential force acting on the same area. Our 
first object will be to determine a, (3, and y, or the position of the base s of the tetra- 
hedron, so that the normal action upon it, p cos shall be a maximum. We shall 
afterwards have a similar investigation with reference to the tangential force p . sin 
* Exercices de Mathematiques, vol. ii. p. 48. 
5 C 
MDCCCLXII. 
