Bootu— On Fresnel’s Wave Surface. 387 
conical point ; the cone would be two planes for some double points. 
Since 
w+ Ard2Ashs = 0, 
then AG = 0, re = 0, A3 = 0, Na = 0 
are the tangential equations of the four real conical points, and 
therefore, from what we have already said, the Cartesian coordi-_ 
nates of these points are written down from, say 
he /@-B + va /P—-C- fae; 
| opm 
that is, eee” 
a — 
as is well known. 
The equation of the tangent cone at a conical point may be 
calculated in various ways, and referred to its vertex as origin, has 
for its equation 
e (7-0) 7 Z (a+ 0) az 
b-—e ee 40°¢ a—b ac J (@- b?) (0? ifr *) a 0, 
the axes are parallel to the axes of the ellipsoid of elasticity. 
Now, since this cone has for its vertex 7,3, and envelopes 
the ellipsoid, 
ee + y + i aes 0 
PE@LB) 288 VIO) Ciro 
its equation is? Bln ey — OF 
and the reader can verify that the equation is equivalent to the 
form just given when the values of ay,2, are substituted. 
1 See Dublin Moderatorship Examination, 1845; Griffin’s Tract on Double Refrac- 
tion ; or Bassett’s Physical Optics, page 123. 
2 P= 0 is the polar plane of x1y121 with respect to = = 0. 
3 Tn connexion with what has been written, the reader might consult the footnote 
on page 438, Art. 405, of Salmon’s Surfaces ; and for an interesting proof that the 
plane 7; = 0 touches the wave surface in a circle, see page 184 of the Solutions of the 
Cambridge Problems aud Riders for 1878, edited by Dr. Glaisher; he ought also to look 
up the footnote on page 273, Art. 192, of Preston’s Theory of Light (first edition). 
