736 
ME. HOPKINS ON THE THEOET OE THE MOTION OE GLACTEES. 
equation expresses a force less than the weight of the whole fluid contained in the tube ; 
and if we call this last weight W,, that equation shows that the entire effect of the 
friction, estimated in the tangential direction in which it acts at each point of the tube, 
is less than Wi. 
Again, let SXi be the part of the friction on any element of the tube, resolved parallel 
to the axis of x ; then 
V'V 7 
0 JVi — J cods . ^9 
similarly, we have 
dx /»*! 7 dz dx 
dz dy 
ds ' ds’ 
• dco dy dz 
Since and ^ are each less than unity, it follows that Xj, Yj, and Zj are each less 
than W, ; and if, taking all the tubes, 
X=X,-|-X,+&c., Y=Yi+Y3+&c., Z=Z,+Z,+&c., W=W,+W,+&c., 
we shall have X, Y, and Z each less than W, the weight of water in the whole mass of 
the glacier. 
The internal pressure Z will tend to elevate the upper surface of the glacier; the 
pressure Y will be equilibrated by the pressures on the opposite flanking walls of the 
glacial valley. To interpret the meaning of X, we may suppose the vertical section of 
the glacier near its origin to be immoveable; then will the pressure X tend to urge 
onwards the whole mass in the direction of the containing valley, where there is no 
barrier to oppose its action. 
To form a conception of the magnitudes of the pressures X, Y, Z, we may remark that 
the water in our supposed tubes must almost necessarily descend through the mass 
• • . • dz 
m approximately vertical directions. Admitting this supposition, ^ will nearly equal 
unity, and Zj will nearly equal / guds^ or nearly the weight of the fluid contained in 
the single tube. Consequently the whole vertical resolved part of the friction=Z=: 
approximately the whole weight of the fluid contained in the glacier. Y will be equal 
only to a small portion of W, because in the case we have taken, will be very small ; 
and X will be also very small, because ^ will be so. And thus it follows that the whole 
internal pressure, arising from the friction on the sides of the small ducts through which 
the water descends, and tending to urge the glacier forwards, cannot exceed a small 
fraction of the weight of the water descending through its pores, and this latter weight 
again must doubtless be an exceedingly small fraction of the weight of the glacier. 
