ME. HOPKINS ON THE THEOET OP THE MOTION OE GLACIEES. 
695 
“ If we take the three values of p deducible from this equation, and substitute them 
successively in equations (c.), those equations combined with ( 2 .) will give three distinct 
systems of values for cos a, cos (3, and cos y, belonging (as is well known) to three lines 
perpendicular to each other. 
17. “ Hence it follows that there is at every point (P) of a continuous solid mass 
under extension or compression, a system of three rectangular axes, such that if the small 
plane s at P be so placed that its normal shall coincide with one of those axes, the whole 
resultant action on s shall be normal to it, the tangential action upon it being then equal 
to zero. These three axes are called the axes of princijpal pressure or tension with refer- 
ence to the point P. 
“ Of the three values oip in these directions, though they all satisfy the conditions of 
maximum or minimum, one is a maximum, another a minimum, and the third is neither 
an absolute maximum nor an absolute minimum. This is best explained, perhaps, by 
converting equation ( 1 .), as CxiUCHT has done, into an equation to a surface of the second 
order, by putting 
^cosS=^, rcosa=.r, rcosj3=y, r cos 7 = 2 . 
The inverse of the square of any radius vector will be a measure of the normal action 
through P perpendicular to this radius vector, the axes of this surface of the second 
order coinciding with the axes of principal tension or pressure. Of the three principal 
axes of this surface, the directions of the greatest and least will manifestly coincide with 
those of minimum and maximum tension ; but though the tension in the direction of the 
mean axis of the above surface satisfies the two conditions and 
it satisfies the one because it is maximum with respect to a, and a minimum with respect 
to fS, or the converse, as the mean axis of an ellipsoid is a maximum in one principal 
section of the surface, and a minimum in the other.” 
18. The object of the second part of this investigation* is to determine the angular 
positions of the small plane (s) passing through P, so that the tangential force acting 
upon it shall be greatest, ^. e. that p sin S may be a maximum. Our formulae will be 
much simplified by taking the axes of principal tension or pressure as the coordinate 
axes. In this case we shall have 
D=0, E=0, F=0; 
and if Aj, B,, Ci now represent the principal tensions at the proposed point, and /3„ y, 
be the values of a, j3, y referred to these new axes, equations (a.) will give 
p'^=A\ cos^a,-l-Bl cos^/3i-f Cf cos^yi, 
and equation ( 1 .) gives 
p . cos^=AiCos®ai-l-Bi cos*/3i + CiCos^y,. 
* A solution of the problem above investigated was also given by M. Cauchy, in his ‘Exercices de 
Mathematiques ’ (vol. ii. p. 48). The solution of the problem in this second part of the investigation has 
only been given, I believe, by myself. 
5 c2 
