704 
ME. HOPKINS ON THE THEOET OF THE MOTION OF GLACIEES. 
The algebraical sum of these pressures, estimated in the positive direction of x. 
dx 
zhxly. 
Again, if W be the weight of a unit of volume of ice. 
Weight of the element 
and its resolved part parallel to the axis of x 
=-'Wz^x^y sin/. 
Also we shall have the tangential force on the base Q,q, 
= — 
parallel to the axis of x. 
Hence we must have for a condition of equilibrium of the element Mg' 
and 
E=(-^+Wsm,)^. 
It would seem probable that A will generally vary slowly with y ; it may vary more 
rapidly with z, especially in cases where it becomes large, as at the bottom of an ice- 
fall. In such case we shall have 
where ^ is taken parallel to z. In the position just mentioned (the foot of an ice-fall j 
dA 
the variation with regard to x may possibly be rapid, and therefore very considerable. 
Under the ordinary conditions of a glacier, away from any rapid fall, the variations of 
rfA 
A must generally be slow, and the values of ^ therefore comparatively small. 
C is manifestly due to the resolved part of the pressure of Mg on the surface Qg 
referred to a unit of surface. The normal pressure on Qg'=W.2:Sa^^y cos /. Whence 
C=W2:cos /. 
This value of C shows that it must always be small when 2 is so. Such will also be 
dh. 
the case with E, unless be very large, which is probably true only at the foot of an 
ice-fall. Generally, then, we see that C and E will be small for all those depths which 
lie within the sphere of our observation ; and that for all such depths the first approxi- 
mate solutions of our general equations are sufiiciently accurate. The second approxi- 
mate solutions give the results for greater depths, and indicate the nature of the results 
for those still greater depths at which C and E might be too large to render the results 
of the second approximation applicable with sufficient exactness. 
We may now distinctly understand the interpretation of the first approximate solu- 
tions of our general equations, as applied to the case of an actual glacier. In those 
