The solution of problems relating to vibrations, in connection with spheroids— or, in two dimensions, elliptic cylinders—has hitherto only been attempted in one manner. If the vibrations have a time factor for their fundamental vector, of the form ℯ
ipt
, the equation of wave motion becomes, if ø is the fundamental vector, (∇
2
+
k
2
) ø = 0 where the wave-length is 2π/
k
,and if C is the velocity of propagation of the wave in the external region,
k = p/c
. When oblate spheroidal co-ordinates are used, defined in terms of Cartesians by x=
a
√ { (1 + μ
2
) (1 + ζ
2
) } cos ω,
y
=
a
√ {1 - μ
2
) (1 + ζ
2
) } sin ω, z =
a
μζ, this can be transformed, after the usual manner, to ∂/∂μ (1 - μ
2
) ∂ø/∂μ + ∂/∂ζ (1+ζ
2
) ∂ø/∂ζ +
k
2
a
2
(μ
2
+ ζ
2
) ø = 0, (1) when there is symmetry round the axis of
z
.
Disordered critical wave functions in random-bond models in two dimensions: Random-lattice fermions atE=0without doubling