216 REPORT—1885. 
which can also be expressed in terms of the principal velocities at right 
angles to the axis, for if w,, w, be the values of these, we have 
Aa? J? 
9 
M+WN=o,? + o,? — 
(8) 
(M = N)? = (w,? = Ww”)? 2 ei 5”) 
The laws obtained in this paper are further developed in a second 
and third in the same journal. In this third paper, Boussinesq! points 
out the necessity of including in the expression for U in terms of u 
differential co-efficients of u,v, w with respect to the time, and shows 
that the phenomena of magnetic rotation can be secoumead for by putting 
in the case of a wave travelling parallel to z— 
yeas Au — 3) 
9 
V = Av + 8 MY (9) 
C 
W= Aw 
while the phenomena presented by refraction at the surface of a moving 
body are explained on the supposition that in finding d?U /dt? we have to 
take into account the visible motion of the body, and write 
d 
d d 
= Le [ — : : , 
7 (Gt, Pa Nz) (10) 
la 
L, M, N being the components of the velocity at the point 2, y, 2; it 
is shown that in cases in which L, M, N are small compared with w’, 
the apparent velocity of light in the body is 
py oe ae 
=w+ — ly. 
B 
# being the refractive index and V the velocity of the body in the 
direction in which the light is travelling. This, of course, is the formula 
given by Fresnel. 
§ 3. M. de St. Venant,? in the article already quoted, sums up his criti- 
cism of the theory as follows: ‘Les deux hypotheses principales de cette 
théorie nouvelle me semblent bien prés de s’élever a la hauteur de choses 
démontrées.’ At the same time there remains the difficulty pointed 
out by Sarrau* of explaining on mechanical principles how the various 
terms in U, V, W arise, and on what physical phenomena the mechanical 
forces brought into action depend. 
§ 4. A further step in the progress of the theory was brought about 
by the discovery of anomalous dispersion by Christiansen‘ in 1870. Le 
1 Boussinesq, Liowville’s Journal, t. xiii. pp. 340, 425. 
? De St. Venant, ‘Sur les diverses méthodes de présenter la théorie des ondes 
lumineuses,’ Ann. de Chimie, t. xxii. 
3 «Théorie des ondes lumineuses,’ Ann. de Chim. (4), t- xxvii. p. 272. 
4 Pogg. Ann. t. 141, p. 479; t. 143, p. 250 
