On Systems of Rays. 133 
single ray-velocity ; and if we take new rectangular co-ordinates x,y, z,, such that 
the plane of «x, z, is still the plane a c of the extreme axes of elasticity, but that the 
positive semi-axis of z, coincides with the line p’, we ney employ the following for- 
mule of a eon 
=X +2005 Y=Yip a —2,pat+Z,Pes (C*) 
which give 
Par tyi +27, =z, 7r”=7, sin. (p p') +2, COs. (¢' 9’), (D") 
and change the equation (4) of the wave to the form 
1=)7 27? +4h2,4 (c?—a@”) sin. ('p’) +4 (0? +.) (7 +”) 
+4 (c°%-a”) Va? +y? V(,sin. (e'p )—2,cos.(op") PP +y”s (E") 
This equation enables us easily to examine the shape of the wave near the end of the 
radius 9’, that is, near the point having for its new co-ordinates 
2 =Ony 012, =); (F”) 
for it takes, near that point, the following approximate form, 
z,=b— 4B Vo? —b— b- Vb — a an (4, + Vx +Y, ay (G") 
which shows that at the point (#"") the wave has a conoidal cusp, and is touched not 
by one determined tangent plane but by a tangent cone of the second degree, repre- 
sented rigorously by the equation (G@"). FresneL does not appear to have been 
aware of the existence of this tangent cone to his wave ; he seems to have thought 
that at the end of a radius 9’ of single ray-velocity, the wave was touched only by two 
right lines, contained in the plane of ac, namely, by the tangents to a certain circle 
and ellipse, the intersections of the wave with that plane: but it is evident from the 
foregoing transformation that every other section of the wave, made by a plane con- 
taining the radius-vector 9’, is touched, at the end of that radius, by two tangent lines, 
contained on the cone (G"). It is evident also that there are four such conoidal 
cusps, at the ends of the four lines of single ray-velocity, +9, +”. They are deter- 
mined by the following co-ordinates, when referred to the axes of elasticity, 
ze: a_h pa a, b2_c2 
t= sgt ei 4 Qe at RV (eae (H") 
and they are the four intersections of FresNet’s circle and ellipse, in the plane of ac, 
which have for their equations in that plane 
Gate, Ot -- oz OC. Ge) 
Again, if we employ the following new formule of transformation, 
‘ ‘ 
DHL Wet Way Y=Y,» Z=—Lweth,wWes (K") 
VOL. XVII. 2M 
