OF THE TRANSLATION OF A DIRECTED MAGNITUDE. 
205 
the displacements represented by v; while fl.v denotes, in magnitude and plane, the 
lateral disarrangement of the medium. 
It is clear, therefore, that the symbolic forms u.v and uxv must be of great use in 
Physical Optics ; indeed the facility they give of following out investigations respect- 
ing undulatory movements is so great, that the whole subject of reflexion and re- 
fraction, in crystallized as well as in uncrystallized media, and the mathematical ex- 
planations of the phenomena connected with polarization, double refraction, &c., may 
be reduced to a state of simplicity which could hardly be expected in such a difficult 
subject. 
(109.) In the paper above alluded to, I obtained also the equation of vibratory 
motion generally, for any crystallized medium, without any of those assumptions 
which mathematicians have found it necessary to make in order to render the inves- 
tigation manageable ; especially, without assuming the vibrations of a plane polarized 
rav to be in the plane of polarization, which appears to me to be a highly objection- 
able assumption. By the aid of the symbolic forms, the general equation of vibratory 
motion, where the transmission of transverse vibrations is possible, is thus expressed : 
Here A,, Ag, A 3 are coefficients of direct elasticity , corresponding to Ain equation ( 1 .); 
and B„ Bj, B 3 , &c. are six coefficients of transverse elasticity, corresponding to B 
in ( 1 ). 
Fresnel’s hypothesis, that the vibrations of a plane polarized ray are perpendicular 
to the plane, makes 
B, = B'i, B 2 = B' 2 , B3=B'3, 
while the hypothesis, that the vibrations are in the plane of polarization, makes 
B,=:B2, B2 = B'3, B3 = B,. 
On the former hypothesis the equation becomes 
while, on the latter, it becomes 
-(Dn.){B,(|-|).+B,(§-|) 8 +B.(|-|)y} (4.) 
For transverse vibrations the rarefaction {Ctxv) (see above) is zero, which further 
simplifies (3.) and (4.). By equating the coefficients of a, (3, 7 in (4.), thus simplified, 
we obtain MacCullagh’s three equations. The equation (3.) coincides in every way 
with Fresnel’s theory. 
