(35), following the receipt in [27], through the procedure of expanding around the

point at infinity described in detail in [39]:

Let O denote the

point at infinity on [E.sup.(n)l.sub.[tau]].

By Proposition 6, if Max{([partial derivative]F/[partial derivative]x)(a, b, 0), ([partial derivative]F/[partial derivative]y) (a, b, 0), ([partial derivative]F/[partial derivative]z)(a, b, 0)} < [epsilon], then (a : b : 0) will be a singular

point at infinity.

Let [OMEGA] [subset] H be a domain whose extended boundary contains the point at infinity and suppose that there exists a real point t [member of] R [intersection] [OMEGA] such that [OMEGA] \ (-[-[infinity], t] (or [OMEGA] \ [t, + [infinity])) is a slice domain.

Let [OMEGA] [subset] H be a domain whose extended boundary contains the point at infinity and such that, for all I [member of] S, [[OMEGA].sub.I] = [OMEGA] [intersection] [L.sub.I] is simply connected.

As shown in Figure 2, beginning with [sigma] = 0 (the Neumann problem), we will allow [sigma] to move along a ray inclined at an angle [theta] to the horizontal toward the

point at infinity. We may then track the trajectory of each Neumann eigenvalue in order to determine whether it eventually approaches a Dirichlet eigenvalue (in which case it will be called an IBC-Dirichlet mode) or migrates to infinity (in which case it will be called a "missing mode").

The Euclidean plane [E.sup.2] = {x = ([x.sub.1], [x.sub.2], [x.sub.3]) [member of] [R.sup.3] | [x.sub.3] = 0} and the

point at infinity 1 compose the boundary at infinity [partial derivative][H.sup.3] of [H.sup.3].

Let [Q.sub.0] = ([X.sub.0],[Y.sub.0]) be the image on (4) of the

point at infinity on (3), then we have [X.sub.0] = 7 + 2[alpha] + 3[[alpha].sup.2] and Y = -17 - 15[alpha] - 8[[alpha].sup.2] where [alpha] = [cube root of (2)].