132 Professor Hamitton’s Third Supplement 
‘pp 
extreme axes of elasticity. Fresnex has shown in like manner that any given nor- 
mal direction corresponds to two unequal normal velocities, except four particular 
directions, which we may call tw’, +w’, and which are determined by the following 
cosines of direction, 
2 §2 2 
= —o g= ia 5 Os = Oz, = ON, jaw Ee E = (OG) 
ce 
and in fact it is easy to establish the following expression for the double value of the 
square of the normal velocity, analogous to the expression (JV), 
w=4(a’? +0) +4 (a —c’) cos. ( (ww) +(ww") ), (Z"*) 
which cannot reduce itself to a single value, unless the sine of (ww) or of (ww’) 
vanishes. FresNEL has given the name of optic axes sometimes to the one and some- 
times to the other of the two sets of directions (X') (Y"*); but to prevent the con- 
fusion which might arise from this double use of a term, we shall, for the present, call 
the set +o, +e", by the longer but more expressive name of the directions or lines 
of single ray-velocity : and similarly we shall call the set +w', +w’, the directions or 
lines of single normal velocity. 
New Properties of Fresnev’s ave. This Wave has Four Conoidal Cusps, at the 
Einds of the Lines of Single Ray-Velocity : it has also Four Circles of Con- 
tact, of which each is contained on a Touching Plane of Single Normal-V elocity. 
The Lines of Single Ray-Velocity may therefore be called Cusp-Rays ; and the 
Lines of Single Normal-Velocity may be called Normals of Circular Contact. 
28. The reasonings of the foregoing number suppose that the axes of co-ordinates 
coincide with the axes of elasticity ; but it is easy to extend the results thus obtained, 
to any other axes of co-ordinates, by the formule of transformation which were given 
in the thirteenth number. We shall content ourselves at present with considering two 
remarkable transformations of this kind, suggested by the two foregoing sets of lines 
of single velocity, which.conduct to some new properties of FRESNEL’ S wave, and to 
some new consequences of his theory. 
The polar equation (JV) of the wave may be put under the form 
=4(c? +a) +4 (ce? —a) Srv" t VP Vert, (A”) 
if we put for abridgment 
r=A'p=20, +20 cs r =A'p=40, +20" 5 (B”) 
so that r’, r", are the projections of the radius-vector » on the directions 9’, p’, of 
