123 
But it has been supposed that the displacement ép is tangential to 
the wave, or perpendicular to »; if therefore we write, 
Tom = wp, or T= et (Gt — wy pw, 
then 7 is at least a vector, even on the principles of Fresnel : while, on 
those of Mac Cullagh and of Neumann, it would have the direction of the 
true displacement, or vibration, within the crystal. And thus, without 
any labour of caleulation, but simply by the expressing of the fundamental 
conceptions of Fresnel’s theory in the Lanevace of Quaternions, Sir 
W. R. H. obtains an Equation of the Index-surface, under the following 
SymsoLicaL Form :— 
0= Se (o7 =p?) pt; (a) 
which is easily transformed into the following :— 
1=Syp(u'—$) "pe. (a’) 
He has also verified, that when he writes, 
d=a'§S.AH+h'S. Pity S.¥7', 
a, B, y, being three rectangular vectors, whereof the lengths are a, 5, ¢, 
an easy quaternion translation enables him to pass from these last forms 
to certain others, although less concise ones, for the equation of the in- 
dex surface, expressed in rectangular co-ordinates; one, at least, of which 
latter forms (he believes) was assigned by Fresnel himself. 
2. To pass next to the Equation of the Wave-surface, let p be the 
vector of that surface; or the vector of Ray-velocity; or simply, the 
Ray-Vecror. It is connected with the index-vector, » (if this last 
vector be supposed to be measured jin the direction of wave-propagation 
ttself, and not in the opposete direction,) by the relations, 
Sep=-1, Spou.=0; 
with which may be combined their easy consequence, 
Szép = 0, 
which assists to express the reciprocity of the two surfaces. Hence, by 
some very wnlaborious (although, perhaps, not obvious) processes, depend- 
ing on the published principles of the Quaternions, and especially on 
those of the Seventh Lecture, but in which it is found to be convenient 
to introduce an auailiary vector, 
v=(w— >)" B, 
(which may be considered to have both geometrical and physical signi- 
fications,) Sir W. R. H. infers that v is perpendicular to p; and also 
that it may be thus expressed as a function thereof :— 
y=($-p?y'p™. 
