690 
ME. HOPKINS ON THE THEOET OF THE MOTION OF GLACIEES. 
partial solution of the problem, in which the dynamical difficulties are evaded. It has 
been already explained that when a solid body is acted on by external forces, it generally 
becomes distorted in form and changed in volume. If the forces be insufficient to over- 
come the cohesion of the mass and to dislocate it, they will of course continue to main- 
tain the body in its state of constraint and distortion, and thus to produce, at different 
points of its interior, pressures and tensions varying both in direction and intensity. 
The immediate object of the first part of the investigations contained in the following 
pages, is the proof of certain propositions respecting these internal pressures and ten- 
sions, and the phenomena resulting from them. The problem thus considered is not a 
dynamical, but a statical one, in which certain results are attainable with the same accu- 
racy as in the simplest mechanical problems ; and such are the only results with which 
we are du’ectly concerned in our present researches. If the distorting forces be suffi- 
ciently increased, the mass will be torn or crushed, as already stated, and will then 
move according to the new conditions imposed upon it in its state of dislocation. This 
constitutes the dynamical part of the problem, but, it must be recollected, it does not 
enter at all into the mathematical part of our own investigations. I have thought it 
necessary to point out this distinction, lest any vague objection resting on an imperfect 
or erroneous conception of the problem before us should exercise an undue influence on 
the mind of the reader. 
Section III. — On tlie Pressures and Tensions at any point of a Solid Mass held in a 
position of constraint by external forces. 
13. We may now proceed to the consideration of the general problem, the object of 
which is to investigate the nature of the internal pressures and tensions at any proposed 
point of a solid mass subjected to the action of impressed forces which slightly distort 
it from the form it would assume when acted on by no external forces at all. It will be 
recollected that these forces are supposed insufficient to destroy the continuity of the 
mass. They maintain it in its distorted form, and must therefore be in equilibrium with 
the internal forces arising from the cohesive power of the mass. 
To explain the nature of the distortion produced in any small element of the mass, 
let us denote by s the area of an indefinitely small plane surface passing through any 
point (P). Generally there will be an action between the particles (M) on one side of 
our small plane, and those (M') on the opposite side. Since s is indefinitely small, we 
may represent by /the whole action of M on M', and suppose its direction to make an 
angle I with the normal to the plane s. Then will 
f cos h and /sin S 
be the normal and tangential actions respectively of M on M'; and 
— / cos h and — / sin ^ 
will be the corresponding actions of M' on M. If the normal force be a pressure, it will 
