692 
ME. HOPKINS ON THE THEOEY OF THE MOTION OF GLACIEES. 
14. “Taking any point (P) of the mass, let it be made the origin of coordinates xyz. 
Let the small plane s be conceived as above to pass through P, and let the forces upon 
it when in the positions specified below be denoted as follows, all being referred to a unit 
of surface. 
“ (1) When a perpendicular to the plane coincides with the axis of let 
fB' parallel to y. 
The normal force = A ; the tangential force = 
C' 
z. 
“ (2) When a perpendicular to the plane coincides with the axis of y, let 
'C" parallel to z. 
The normal force =B ; the tangential force — 
M 
X. 
“(3) When a perpendicular to the plane coincides with the axis of z, let 
The normal force =0; the tangential force 
“ Between the six accented quantities there are three essential relations, which are 
easily found. On the three coordinate axes at P, construct an indefinitely small paral- 
lelepiped whose edges are ^x^ ^y^ Iz. The six equations of equilibrium of this element 
will express the conditions that the sums of all the resolved parts of the forces parallel 
to the coordinate axes shall respectively be equal to zero ; and that the moments of the 
forces with reference to these axes shall also severally be equal to zero. Let us take 
the three latter conditions, lines through the centre of gravity of the element and parallel 
to the coordinate axes being taken for the axes of the component couples. The tangen- 
tial force parallel to the axis of x on the side lx.'hz being A', that on the opposite side 
will be — ; and the couple resulting from these forces about the axis 
parallel to z, will be 
Mlxlz. ^A'+ ^ ^xlz . ^ ; 
or, omitting small terms of the fourth order, 
“ Similarly, the couple arising from the forces B' and about the same axis 
parallel to z, will be 
“ Also the moments of the normal forces A, B, C, with reference to the above-mentioned 
axes, will be zero, always omitting small quantities of the fourth order. Consequently 
the whole moment of the forces on the parallelepiped with reference to the axis parallel 
to that of z, will be 
(A'— B')S^^j/§z, 
which must = zero by the conditions of equilibrium ; and therefore 
A'=B'. 
