(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]].

The experience of reading the Essais evokes a point at infinity too.

Its capacity to generate movement toward a point at infinity would have suited a king invested in the notion of "the king's two bodies," (20) whereby as "the king" approaches death (at which point one will say "le roi est mort"), "the king" does not approach death (and so one will also say "vive le roi").

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)].