172 REPORT—-1885. 
For an isotropic solid we should have N,’=N;'=(u#—2L)e and 
‘T,/=0. Thus Green’s medium in which the propagation of transverse 
waves is possible has properties which distinguish it from an isotropic 
solid, for a simple elongation produces on any plane parallel to the direc- 
tion of the elongation a normal stress which depends on the position of the 
plane, while it also produces shearing stress about an axis parallel to the 
direction of the elongation; and although the theory does not explain 
double refraction satisfactorily, yet it is not open to De St. Venant’s criti- 
cisms on this point. 
§ 5 In the same paper St. Venant proposes a modification of Cauchy’s 
theory which leads to Fresnel’s wave surface without any more conditions 
than are required by Green; for, putting in Green’s expression, 
ApS Ep? + Gn? =X | 
l, m, n being the direction cosines of the wave normal, the equation to 
determine the velocity becomes— 
{po V2 — X — Gl? — Hm? — In} [(p V? — X)? — (p V2 —X) 
{M+N)2? +(N+L)m? 4+ (1+ M) 07} + MN? + NUIm? + LMn?] 
— {((H—L) (I— L) — (LU + P)} {G22 + Nm? + Mr? + X — pV?} mn? 
— {(I—M) (G—M) — (M+Q)*} {N2? + Hm? + Ln? + X— pV} v7? 
—{(@—N) (H—N) — (N+ B)?} (MP? + Lm? + In? + X — pV?} Pm? 
+ {(G—M) (H—N) (I-—L) + (G—N) (H-L) (I—M) 
—2(L+P) (M+Q) (N+ R)} Pm? =0 . y Y ; . (4 
And this will reduce to Fresnel’s surface if A = B= OC; that is, if the 
equilibrium stresses are equal, and the four conditions 
(H—L) (IL) =(L+P)? 
(I—M) (@—~ M)=(M+ Q)? 
(G—N) (H-N)=(N+R) - (6) 
(G—M) (H—N) (1—-L) + (@—N) (H-L) d-—) 
—2(L4+P) (M+ Q) (N+R) =0 
are satisfied. 
These equations include those of Green’s first theory, and are approxi- 
mately those which arise from what. St. Venant calls an ellipsoidal 
distribution of elasticities. Under certain circumstances the tasinomic 
surface—which, it will be remembered, gives the tension in any direction 
produced by a simple elongation in that direction—reduces to an ellipsoid, 
and then the distribution of elastic constants is named by St. Venant 
ellipsoidal. This distribution is produced when an isotropic medium is 
unequally strained in three perpendicular directions. The theory is 
interesting, and important as showing that Fresnel’s wave surface can 
be deduced from the general elastic solid theory on other assumptions as 
regards the constants than those given by Green, and that the vibrations 
in this case are not necessarily in the wave front. There will, however, 
in this case be a quasi-normal wave, the velocity of which is given by the 
equation 
pV? — X— GP? — Hm? — In? = 0; 
