474 PROFESSOR C. NIVEN ON THE 

From the formulee which determine the small motions of such a substance, 
when strained homogeneously at first, it follows that the vibrations in a plane 
wave are not generally in its front, and that for each position of the latter there 
are three real and different velocities of transmission. Ifthe primary stress be 
symmetrical with respect to an axis, the wave surface breaks up into an 
ellipsoid of revolution and a surface of the fourth class.* 
In these investigations it was necessary to determine expressions for the 
stresses due to distortions of any magnitude. My results for these, though not 
their symbolical form, have, as I find, been already published in the ‘Comptes 
Rendus,” t. lxxi. p. 400, by M. Boussinesg. In the present paper two demon- 
strations are offered, one derived from the rules for compounding two strains. 
The. present paper contains also a general theory of the laws according to 
which strains and certain other physical magnitudes are transformed with 
respect to different sets of rectangular axes. 
§ 2. Let the three intersecting edges of a rectangular element-parallelopiped 
PH, PK, PL, be called h, &, 7, and let them be strained into P’H’, P’K’, P'V’, 
the displacements of P being w, v, w. The co-ordinates of H’, K’, L’, relative to 
P’ are h, &, J, multiplied respectively by the members of the successive columns 







WU, Uy, Us 
of the determinant | 7, v, v, | = V, 
W, W, Wy 
d du a __ du =. dw 
where u=7, (e+uj=14+7, Ug = Get U) =a ...W,=1+5. 
The minors of its constituents wu, ...w, we denote by U,... W,; and the 
geometrical elements of the strain are expressed by these laws :— 
(1) Any infinitesimal volume, dV, strains into dV’, where dV’= V.dV. 
(2) If any infinitesimal surface-element d=, whose projections on the co- 
ordinate planes are A,, A,, A,, strains into d>’ having corresponding projections 
Ay, Aj, Aj, then 
A,=A,.U, +A,.U, +A,U, 
A,=A,V, +A,V, +A,V, : : (2). 
A;=A, W,+A, W,+A, W, 
(3). The six sirains are given by half the sums of the squares of the elements 
in the 3 columns of V less 1, and by the sums of the products of the elements 
. 
: If the solid be ‘‘ incompressible” for small strains, and if the primary strains be small, the equation 
giving the velocity of transmission of a plane wave is similar in form to that given by FRESNEL in 
his Theory of Double Refraction. 
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