696 
ME. HOPKINS ON THE THEOET OF THE MOTION OF GLACIEES. 
Hence we have (if sin ^=T) 
sin^^=Ai cos'^ ai + B, cos^/3i4-Ci cos^ — (Aj cos^ Ki+Bj cos^/3i+Ci cos^ 
which is to be a maximum subject to the condition 
cos^ a, + cos^ jSj + cos® y 1 = 1 . 
The solution of this problem is longer and more complicated than that of the problem 
solved in the first part of the investigation. It will here be sutficient for my purpose 
to state the results, and, for the analytical solution, to refer the reader to the volume of 
the Cambridge Transactions above quoted. The results are as follows : — 
a„ |3i, and yi being the angles which define the position of the small plane s (art. 13), 
the analytical conditions for the tangential force (T, or jp sin upon it being a maxi- 
mum or minimum, are satisfied by the following three systems of contemporaneous 
(1) a, = 90, /3i=yi= + 45, 
(2) /3,=90, y, = «,= + 45, 
(3) yi = 90, ai=^,= +45; 
(d.) 
and if T,, T 2 , and T 3 be the values of T corresponding respectively to these systems of 
values of a„ | 8 ,, and yj, we have 
T®=i(B.-C0®, T|=i(A,-C0®, T®=i(A,-BJ®. 
If A,, B,, Cl be taken, as they always may be, in order of magnitude, Tg will mani- 
festly be the greatest of these values of T. It is in fact, as shown in the memoir 
referred to, the only value which satisfies all the conditions of a maximum. The corre- 
sponding values of a,, /Bi, and yi, which determine the corresponding position of the 
plane s, are those given by the second system of (d.). Now (3^ is the angle between the 
normal to s and the axis of j/; and since it is here =90°, the normal to s must lie in 
the plane of xz, and the plane s itself must pass through the axis of y. Moreover, 
since the corresponding values of and yi are each +45°, this plane may have two 
positions, in both of which it bisects the angle between the coordinate planes of and 
yz, these positions being on opposite sides of the plane of ^z. These considerations, 
however, only determine the position of the plane in which the maximum tangential 
force (T 2 ) acts ; they do not determine the linear direction of the force in that plane. 
It is easily shown that it is perpendicular to the axis of y *. Since we have here D=0, 
E = 0, and F=0 (art. 15), we have from equations (a.), 
]) cos \ =Ai cos a,, 
'p cos(jj =Bi cos /3i, 
p cost- =Ci cos yi, 
p being the whole resultant force on the small plane s, referred to a unit of surface, and 
* No proof of this is given in my memoir above referred to in the Cambridge Transactions. 
