ME. HOPKINS ON THE THEOET OE THE MOTION OE OLACIEES. 
699 
This corresponds to two values of cs differing by 90°, These two values of a, and y=90° 
(since cosy=:0), determine the two positions of the axes of the principal tensions j?, and 
^ 2 - They both lie in the plane of xy. 
Taking the third value ofy>=^ 3 — 0, the equations (c'.) are also satisfied by 
cosa=0, cos|3=:0; 
by which the equation 
cos® a + cos® (3 + cos® 7 = 1 
is reduced to 
cosy= + l. 
These values of a, j3, and y correspond to the axis of z, showing that axis along which 
the pressure =0 to be the third axis of principal tension. 
21. In determining the magnitude and direction of Tj, the greatest tangential action, 
we must bear in mind the remarks in art, 19. The quantities denoted in the preceding 
formulae by y)i, jjg? the same as those denoted in art. 18 by Aj, Bj, Cp The former 
are here retained for greater distinctness. If A+B be >M, ^2 wi^ be positive, and the 
order of the principal tensions will be Pi-, p 3 - The axis of ^ will then be the mean 
axis, and the direction of the maximum tangential action, Tg, will bisect the angle 
between the other principal axes of x and z. Also we shall have 
=i{A+B+M}.J 
If, on the contrary, A+B be <M, yjj will be negative, and the order of the principal 
tensions will be y>i, p^, p^. The direction of Ta will be perpendicular to the axis of z, 
bisecting the angle between the principal axes of x and y. Also we shall have 
These two cases hold when 
=:M. 
respectively, or 
M<or>A+B 
(A - B)® + 4F® < or > ( A + B)®, | 
F® < or > AB ; J 
if) 
(N.) 
or the second case may hold when A and B are both tensions, or both pressures, pro- 
vided one of them be small ; and it must necessarily hold when one is a pressure and 
the other a tension. 
22. Solution of the General Equations to a Second Approximation. — In proceeding to 
second approximation I include C and E, but regard their magnitudes as small. These 
magnitudes must be expressed by the ratios which they bear to some standard force. 
The greatest value which the tangential force F can attain in any glacier must be limited 
by the tangential cohesive power of glacial ice ; for if F exceeded this latter force, dislo- 
cation, by a tangential sliding of one element past another, would instantly ensue. Let 
this tangential cohesion be measured by Fij then, when it is said that C and E are small, 
