ME. HOPKINS ON THE THEOEY OF THE MOTION OF GLACIBES. 
735 
atmosphere, and such, therefore, as can exert any influence in expanding the mass, and 
promoting its onward motion. It will act on the interior of the mass in directions 
normal to the tubes. But there is another force also, friction, acting on the mass of the 
glacier in this case, in directions tangential to the tubes, the magnitude of which requires 
to be considered. It retards the motion of the fluid, acting equally on the fluid and 
on the sides of the tubes containing it in opposite directions. Now in the case before 
us, ^=^ 1 , the atmospheric pressure, and, therefore, by equation (1.), art, 61, 
J' fds—g{z—z,)—l{v^—vX)', 
and difierentiating. 
and 
^ dz ^ dv^ 
( 2 .) 
where^y^ fcads is the whole retarding force of friction in any one of the tubes whose 
length =5, (art. 61), and is therefore the amount of friction produced by the water on 
the sides of the whole tube. 
To find the value of the last term in equation (2.), we have (the duct being always 
full) 
(3.) 
where is the value of a when 5=0 and v=-v^. Therefore 
2 ^ 
^ 2 ’ 
dv'^ 2v^co^ dui 
ds 00^ ds' 
and 
Now we may venture to assert that when water descends through exceedingly minute 
ducts in the manner here supposed, the velocity with which it will permeate the lower 
portions of the mass will be much the same as that with which it will pass through the 
upper portion. Assuming this, we shall have by equation (3.) and the value of 
the above definite integral will =0. Hence, by equation (2.), we have 
guds is the weight of an element of the water in the tube, and gcuds.— is a force less 
than that weight. Consequently the definite integral on the right-hand side of the last 
5 H 2 
