ME. HOPKINS ON THE THEOET OF THE MOTION OF GLACIEES. 
701 
If we were to substitute these expressions in the first equation, we should obtain the 
cubic in p used in the immediately preceding articles for the approximate determination 
of its value. Substituting the values of cos a and cos /3 in the equation 
cos^a + cos^/3 + cos^y=:l, 
we obtain 
or 
or 
The retention of in the coefficient of cos^ y would only produce a term, in the expres- 
sion for cos^ y of the order E^, and may therefore be neglected. Also, since p only dif- 
fers from its first approximate value, p^, by a small quantity of the order E*, we may, 
for the reason just assigned, write for^ in the last equation. Thus we have 
and by substitution. 
1 
These equations show that y is changed from a right angle to one whose difference from 
a right angle is of the order E, and therefore small ; while cos a and cos (3 are changed 
by small quantities of the same order as the difference between p and p^, i. e. of the 
order E^ 
24. Nature of the Forces A, B, C, D, E, and F in the ordinary cases of Olaciers . — In 
the preceding investigations we have considered the forces A, B, C, D, E and F as 
acting on any element of a solid body. In the case of a glacier, this body assumes a 
specific form, and it becomes necessary to explain what forces are represented by the 
above symbols in this particular and restricted case. The phenomena with which we 
shall he here concerned, have been observed almost entirely in those regions of glaciers 
in which most large ones, like those of the Alps, become much elongated in consequence 
of the narrowness of the valleys down which they descend. The sides of these valleys 
frequently approximate to parallelism with each other. The primary general charac- 
teristics of the motion of glaciers of this kind are (1) the motion is unaccelerated, 
(2) the axial portions move with a greater velocity than the marginal portions, and 
(3) the superficial portions move somewhat faster than the lower portions of the mass. 
These points are clearly established hy observation, independently of any particular 
theory. In the following articles of this section their truth is assumed. 
MDCCCLXII. 5 D 
