10 
ME. MACQIJOEX EAXEESTE ON THE STABILITT OF LOOSE EAETHL 
That is to say, symbolically, let E be the total pressure, per unit of area, at any point 
of the given plane, making with the normal to the plane the angle of obliquity 6-, let P 
be the normal and Q the tangential component of E ; so that 
P=Ecos^; Q=Esin^; 
-p=tan^; 
then it is necessary to stability that 
Q<F=P tan <p, 1 
and consequently that (2.) 
§ 3. Lemmata as to the Composition of the Stress at a point. 
It is well known that the stress at any point in a solid medium is capable of being 
resolved, with reference to any set of three rectangular axes, into six elements, viz. three 
normal pressures, P^, P^,, P^,, on unity of area of the three coordinate planes, and three 
tangential pressures, Q^, Q^, Q^, on unity of area of the three pairs of coordinate planes 
parallel to the three axes respectively. It is also known, that if we take these six 
elementary stresses for the coefficients of what, in Mr. Cayley’s nomenclatme is called 
a Ternary Quadric, and in the nomenclature of a paper on Axes of Elasticity, a Tasime- 
tric Function f , 
\]—T,x^-\-Ypf-\-V^z^-\-2Qpjz-\-2Q,yZX-\-2Q,xy, (3.) 
then if this quadric be transformed so as to be referred to new axes, the coefficients of 
the transformed quadric will be the elementary stresses referred to the new axes ; and 
further, that there is a set of three rectangular axes, being the principal axes of the sur- 
face U=I, for which the tangential stresses vanish, and the normal stresses become 
maxima or minima, the quadric being reduced to 
U = P,^+P^;y^+P,z^ (4.) 
The normal stresses for those principal axes of pressure are called the principal pressures. 
Let P^ be the greatest and P^, the least of the three principal pressmes at a given point 
O, and let On, making with Ox the angle xOn=-ip, be a line in the plane xy. Let E„ be 
the total pressure on unity of area of a plane normal to On, and let the dii-ection of tliis 
pressure make with On the angle ^ on the side of On towards x, so that the components 
of E„ are respectively, normal, P„=E„ cos ^ ; tangential, Q„=E„ sin 
Let the half-sum of the greatest and least principal pressui’es be denoted by 
P 4-P 
and their half-difference by 
D_ P^ Py . 
2 
* Philosophical Transactions for 1854-55, 
On Quantics.” 
t Ibid. 1855. 
