crystalline Reflexion and Refraction. 23 
formation of the coordinates themselves. The like will be true of the quantities 
X,» Y,» Z,» if we put 
dy seta panies, GS, wae, de 
Lo Diake To Umen yaaa antag Wag 7 a 
and so on successively. 
It is to be observed that, in this Lemma, the displacement is not limited by 
any restriction whatever. Each of its components may be any function of the 
coordinates. But the displacements produced by a system of plane waves are 
restricted by our definition of such waves; they must be the same for all par- 
ticles situated in the same wave-plane. If the waves be parallel, for instance, to 
the plane of 2’ y’, the quantities ’, »/, ¢ will be independent of the coordinates 
a’, y’, and will be functions of z only. This consideration reduces formule (p) 
to the following, 
_ oy dE oc af 
X= 77 COS a — TF cos a’, 
__ dy dé 
¥ = Gy 8B — 77 00s B; (E) 
Sd wet? 
in which it is remarkable that the normal displacement ¢’ does not appear. If 
& = 0, these formule become 
di dy dy 
X= 778% Y= dy °O8 Ps z= Fo cosy; (F) 
or if »/ = 0, then we have 
= dé’ 3 ; 
x= — Tz 8” Y= de 8 Bs A a CORY» (¥’) 
Lemma Ill. If, in an ellipsoid whose semiaxes are equal to a, 0, c, there be 
two rectangular diameters, one making with the semiaxes the angles a, p, y, and 
the other the angles a’, p’, 7, such as to satisfy the condition 
cosacosa’ cos cos 
eo et 
cos y cos + 
es = (()) (G) 
