382 Scientific Proceedings, Royal Dublin Society. 
conic is a double line on the surface. The latter case is impossible, 
for the well-known trace of the surface on the plane y=0 shows 
no double points where the plane ,=0 meets y=0; hence the 
plane 7,=0 touches the surface. Again, the etieord P=0 and 
the ellipsoid of elasticity have manifestly the same planes of 
circular section, and therefore ; 
i= 0: ip => a) 1,=0 
are four planes of circular section of the ellipsoid P=0. This 
proves one of Hamilton’s theorems; but it appears that these four 
circles all lie on one ellipsoid, and assuming for a moment that the 
reciprocal of the wave surface is the wave surface of 
BpeuSais! 
(the sphere of reciprocation being «2 +y?+s°-1=0); then, by 
reciprocation, it appears that the wave surface has four conical 
points, the tangent cones at which all envelop one ellipsoid whose 
equation is 
ee 2 y i 3 1 
O(a 20?) § 2a8es a Oieic) ea: 
I shail show later on. 
The reader will have no difficulty in remembering the value of 
P if he recollects that P is the quadric factor of - 
Similarly the equation may be written 
Q’ + mm msm, = 0, 
where Q stands for 
CE +DY +2 +E(e+y +3) -e(a +0’), 
a /o—e + yf eb + Cf Fa0?. 
It is not necessary to write down the values of 2, m3, m. It may 
be noted that all the imaginary sections in this case lie on the real 
ellipsoid Q=0. 
and 7, is 
