1901-2.] 
Lord Kelvin on Stress and Strain. 
99 
elastic solid of the most general possible kind according to Green’s 
theory, expressed in terms of the new mode of specifying stresses 
and strains, without restriction to infinitely small strains. 
§ 4. To understand thoroughly the state of strain specified by 
(1) or (2), let the tetrahedron of reference, A 0 B 0 C 0 D 0 , for the 
condition of zero strain and stress, be equilateral (that is to 
say, according to the notation of § 2 (1) let J of each edge = 
Po ~ % ~ r o ~Po = $o = r o) m AqBoCoDo inscribe a spherical sur- 
face touching each of the six edges. Its centre must be at K 0 , 
the centre of the tetrahedron ; and the points of contact must be 
the middle points of the edges. Alter the solid by homogeneous 
strain * to the condition (p, q , r, p , q , r) in which A 0 B 0 C 0 D 0 
becomes ABCD. The inscribed spherical surface becomes an 
ellipsoid having its centre at K, the centre of ABCD, and touch- 
ing its six edges at their middle points.! This ellipsoid shows 
fully and clearly the state of strain specified by p , q, r, p ', q', r'. 
It is what is called the “strain ellipsoid.” J 
§ 5. Two ways of finding the ellipsoid touching the six edges 
of a tetrahedron are obvious. (1) Through AB and CD draw 
planes respectively parallel to CD and AB; and deal similarly 
with the two other pairs of non-intersecting edges. The three 
pairs of parallel planes thus found constitute a hexahedron which 
contains the required ellipsoid touching the six faces at their 
centres; or (2) draw AK, BK, CK, DK, and produce to equal 
distances KA', KB', KC', KD' beyond K. We thus find four 
points, A', B', C', D', which, with A, B, C, D, are the eight 
corners of the hexahedron which we found by construction (1). 
A circumscribed hexahedron being thus given, the principal axes 
of the ellipsoid, and their orientation, are found by the solution of 
a cubic equation. 
§ 6. Another way of finding the strain- ellipsoid, which is in 
some respects simpler, and which has the advantage that in its 
construction it does not take us outside the boundary of our 
* Thomson and Tait’s Natural Philosophy , § 155 ; Elements , § 136. 
t Thus we have an interesting theorem in the geometry of the tetra- 
hedron If an ellipsoid touching the edges of a tetrahedron has its centre 
at the centre of the tetrahedron, the points of contact are at the middles of 
the edges. 
X Thomson and Tait’s Natural Philosophy , § 160; Elements , § 141. 
