Bootu—On Fresnel’s Wave Surface and Surfaces related thereto. 211 
see “ Surfaces,” Art. 202, Ex. 2. If we had defined the parallel surface as the 
envelope of the sphere 
(c-—avfP+(y—yl+(2e-2)P =F, where - +* 
the application of the same method leads without trouble to the solution as equating 
to zero the discriminant of 
A4a?a?. 4677? 4¢? 2" 
xs 2 2 Cy ae) 
uot op—otae got 2 + y?+2— hk’). (12) 
This equation is equivalent to (11), but it presents the parallel surface to us in an 
interesting way, for if, following Mr. W. Roberts (Salmon’s ‘ Surfaces,” Art. 518), 
we put #+7?+2=/", and then change z into 2/2, y into y/2, and z into z/ 2, 
we have Professor Cayley’s equation (6) given above for the negative pedal. 
The determination of the parallel surface to the ‘surface of elasticity ” may 
also be effected in the following way. Find the envelope of 
Qa!’ + Qyy' +22 =P 47 4+2—h +a? +y? +2”, 
where (2? + y? + P= ea? t+ Py? +e 2”. 
Differentiating we have a—a =)a' (20? — a’), 
y —y = (20° — 6), 
2—2 =z (2t? —c’) 
[where ¢?=2? +4 y?+4+ 2]. 
Also put M=e7+y4+27—h. Hence we'+ yy + 22e— t?= 00", 
— f? 
which is the same as 2\t* = M—t?, Hence \= ae 
»__y (20? — a’) (M2) Ne ian ey ei 
ee St = Pe of and 7 = Figo = aitoy te ait 
and now substituting for 2’ in the given equations, we have the two equations 
47° 47? ieee 42? 
C+2M—v76M vy PLOIM—RPOM  ?&+2M— COM 
=0M+1, (18) 
4a? a? Fe 40? 7? ; 4c? 2? » 
(@+2M—@OM) * (8 +2M—POM) * (2+ 2M—cOMy 
and ils (14) 
Now the second is the derived of the first with reference to @. Hence the 
answer is the discriminant of (13) equated to zero. Now if in (13) we put 
P=e2+y +2", that is, M=0, and change z into z/2, y into y/2, and z into 2/2, 
we get the ellipsoid as we ought. We could ot have done likewise with the 
