698 
]VIE. HOPKINS ON THE THEOEY OE THE MOTION OE GLACIEES. 
and the axis of z will become the mean axis. Other cases must be treated in a similar 
manner, always preserving the order of algebraical magnitudes for determining the 
mean axis. The line of greatest tangential action is always perpendicular to it, and 
bisects the angle between the other two axes of maximum and minimum tension. In 
determining the magnitude Tg, we must take the same precaution, in arranging the 
principal pressures in their proper order of magnitude, to determine which are alge- 
braically the greatest and least. Thus in the above case, where the order is Bj, Cj, — Aj, 
we have T 2 =^(B,-|-Ai). We may remark that the sign of Tg is of no importance in any 
application we are contemplating of these formulae. 
20. Solution to a First Approximation . — The complete solution of the preceding equa- 
tions cannot be generally obtained. For their practical application they must be solved 
by approximation, when the approximate solution may be sufficient. The most import- 
ant case is one in which the problem can be completely solved in consequence of its 
simplification arising from the particular conditions assumed. The case is that in which 
we suppose no forces to act at any point parallel to one of the coordinate axes, as that 
of z. In such case C=0. Also we assume the absence of any couple tending to twist a 
proposed element about the axis of x, or that of ^. e. D=0, and E— 0. Hence the 
equations (c.) (art. 16) become 
(A— p) cosa+Fcos/3=0, 
Fcos a-|-(B— cos j3=0, > [d.) 
( —p)cosy=0; 
and the cubic for the determination ofp becomes 
-j- {(A—p)(B—p)F‘^}p=0. 
This last equation gives for the value ofp, 
i)=i{A+B+^/(A-B)*+4P}, 
^= 0 . 
Or putting (A— B)®-i-4F^=M^, 
p,=i{A+B+M}, ' 
i^2=i{A-l-B — M}, > (d'.) 
P3 = 0. 
For the values p^ andp^ ofp, the third of the above equations (o'.) gives cosy=0; 
and the equation 
cos^ a -h cos^ /3 cos^ y = 1 
gives 
cos /3=sin a ; 
and eliminating p and /3 from the two first of equations {c'.), we obtain 
•.■■•.((o 
tan 2a: 
A-B 
