210 Bootu—On Fresnel’s Wave Surface and Surfaces related thereto. 
and & the normal distance between the parallel surfaces, or find the envelope of 
P+yte2—ar' —yy —z=ht+k, (8) 
where RSA i il. 
a ab: are 
Hence we get - hea! dat’ 
ae ae 
ky’ dy’ 
y+ e > ye? 
kz de 
Anse 
Hence wet yyt+ed@tht=jxHzH4+/V4+2-F; 
eh lh 
2 aa &e. 
CT 
Substituting in (8) we get 
2 pnd B77? Ce ‘ 
a at ro ce Sl 
A—-S 
we ay? by? ee ke > 5 ‘ j ss 
Bleage or Toe Pangan ge oe Ye meee (9) 
Again, if we substitutein ?=2?+y"+ 2", we get 
ax bty? ee" ot ke 
(i+ oe" * + eryt (+ecy @-% ee) 
which is the derived equation of (9) with regard to 6. Hence the parallel surface 
is obtained by equating to zero the discriminant of (9) considered as a quartic in 0, 
and the application of the method employed in Salmon’s “Surfaces,” Art. 206, to 
the equation (9), reduces to a definite algebraic problem the determination of the 
‘““surface of centres” of the surface of elasticity. Again, the same method gives us 
the parallel surface to an ellipsoid (without using the Calculus of Invariants and 
Covariants), in fact, this is the same thing as the determination of the envelope of 
(xa + yy +22 —kPa=oeu?t+ By? +022, where 2?+y?+2?=1. 
Proceeding precisely as before, it will be seen without trouble that the result is 
obtained by equating to zero the discriminant of 
a ey? 2 Ie 
a ey ke a Se 11 
regarded as a quartic in 6. This is the equation obtained otherwise by Salmon, 
