ON OPTICAL THEORIES. VFI 
efficient remaining a finite quantity, and in consequence the results are 
erroneous. 
§ 2. Lamé is the author of numerous papers, in the ‘Comptes 
Rendus’ and elsewhere, on the propagation of waves through an elastic 
medium, and his results are summed up in his ‘ Legons sur |’ Elasticité.’ ! 
The general form of the equations for the strains are shown to contain 
twelve constants, which become six if the dilatations be equated to zero, 
and three when planes of symmetry are taken for the co-ordinate planes. 
The equations of motion finally obtained may be written 
du — 2 du _ | _ 72 d (dw _ “) (1) 
dt? dy ei da rats dz , ; 
etc., which agree with MacCullagh’s and with Green’s if we omit the 
terms involving the dilatation. The arguments to be advanced against 
the theory are identical, then, with those which Professor Stokes has urged 
against MacCullagh’s. 
§ 3. St. Venant has written many most important papers on the 
subject of elasticity. He still adheres to Cauchy’s theory and the form of 
the equations of an elastic solid deduced from the hypothesis of direct 
action between the molecules of the medium, and in his last great work 
on the subject, the annotated French edition of Clebsch’s ‘ Elasticity,’ 
states his reasons for so doing in §$ 11, 16. However, in the work he 
employs Green’s expression for the energy, with the twenty-one co- 
efficients—‘ Vu la controverse actuelle ot la majorité des avis est con- 
traire au ndtre.’ 
§ 4. In & paper printed in 1863? he criticises Green’s theory of double 
refraction, arguing that Green’s conditions for the tranversality of the 
vibrations lead to isotropy. This conclusion is frequently repeated in St. 
Venant’s* papers, and it will therefore be well to investigate the point 
somewhat closely. 
Let us suppose that we have a simple elongation « in a direction 
1, ™,, %,,in a medium fulfilling Green’s conditions. Let 1, ms, no, 1s, 
m3, 3 be the direction cosines of two lines at right angles in a plane 
normal to 1,, m,, 7, and let us investigate the stresses N,’, N,’, N,’, 
T,’, T,’, T;’ on the faces of an element normal to these directions. Then 
St. Venant’s argument rests on the fact that N,’ is independent of the 
direction of the elongation, while T,’and T,’ vanish, and that this would 
be the case in an isotropic medium. This last statement is of course 
true, but on Green’s theory N,’, N;' do depend on the direction, which 
they would not do in an isotropic medium, and T,’ has a finite value, 
while for an isotropic medium it would vanish. 
The values for the stresses may be shown to be— 
ye 
N,! = in B(L int +Xn)} 
Nz! = {u—2(L1,? + Mm,?+ Nn,”)} «| : F (2) 
T,’ = 2 {Li,l3 + Mmm; + Nirgns} | 
Le == Ts’ = 0 
1 Lamé, Lecons sur ? Elasticité. Paris: Gauthier Villars, 1866. 
? St. Venant, ‘Sur la distribution des élasticités autour de chaque point d’un 
solide,’ Liowville’s Journal, 8. ii. t. viii. p. 257. 
4 . a De St. Venaut, ‘Théorie des ondes lumineuses,’ Ann. de Chim. 
- iv. p. 22, 
