100 
Proceedings of Royal Society of Edinburgh. [sess. 
fundamental tetrahedron, is as follows: — In the equilateral tetra- 
hedron A 0 B 0 C 0 D 0 describe, from its centre K 0 , a spherical surface 
touching any three of its faces. It touches these faces at their 
centres ; and it also touches the fourth face, and at its centre. 
Hence, if we solve the determinate, one-solutional, problem to 
draw an ellipsoid touching at their centres any three of the four 
faces of any tetrahedron ABCD, and having its centre at K, this 
ellipsoid touches at its centre the fourth face of the tetrahedron ; 
and it is the strain ellipsoid for the homogeneous strain by 
which an equilateral tetrahedron of solid is altered to the figure 
ABCD. 
§ 7. To bring our new method of specifying strain and stress 
into relation with the ordinary method for infinitesimal strains 
and the corresponding stresses : — Let X denote the length of each 
edge of the equilateral tetrahedron of reference, A 0 B 0 C 0 D 0 ; and 
let h be the edge of the cube of which A 0 , B 0 , C 0 , D 0 are four 
corners (this cube being the hexahedron found by applying either 
of the constructions of § 5 to the tetrahedron A 0 B 0 C 0 D 0 ). The 
twelve face-diagonals of this cube are each equal to X, and there- 
fore X = h J 2. Let now the cube be infinitesimally strained so that 
its edges become 7i(l + e), 7i(l + /), h( 1 + g) ; and so that the angles 
in its three pairs of faces are altered from right angles to acute and 
obtuse angles differing respectively by a , b, c from right angles. 
This is the strain (e, /, g, a , b, c) in the notation of Thomson and 
Tait referred to in the introductory paragraph above. By the 
infinitesimal geometry of the affair, we easily find the correspond- 
ing alterations of the face diagonals, which according to our present 
notation are (p - 1)A, {p - 1)A, (q - 1)A, etc., and thus we have as 
follows: — 
V - 1= K/+!7 + “) 
P - ^=W+9-a) 
q-1 =^g+e + b)\ 
g'-l=i (g+e-l) W 
r - 1 *= -|( e + /+ c) 
r -l=±(e+f-c) , 
for the relation between, the two specifications of any infinitesimal 
strain. Adding these, and denoting e +/ + g by s, we find 
p+p +q + q' + r-\-r' - Q = 2 s .... (6). 
