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By W. E. Kirwan, L. Zalcman

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T 2 = x ~+I/R2 Rkp~l)(t ) is clearly Similarly a homogeneous if k Rk_(~ r k )(t) where ~ homogeneous is a polynomial polynomial It remains x = Xm+ I, : = is a polynomial of Thus 89 cv x2vm+IRk-2v ~=0 of degree k in xI to Xm+ I. k = 2p + i, RtR2P~(t 2) of degree of degree to show that p2 = [mv:l xg, 2 P~I) by Lemma 4. polynomial is odd, Then v k = Xm+iR2Pr p, so again in xI is harmonic. t : x/R, Q = Rkp(x/R) to Rkp~l)(t) is a Xm+ I. To see this, we write 36 where P = P~k). 7) ~-~ P(t) and for 9 : !

G r a d i e n t of y Results related Rm be a smooth domain on the unit sphere ~n D and i8 p o s i t i v e three [2]. be a smooth f u n c t i o n on the closure of the boundary of The [9]. The main tools of the proof of Theorem lemmas. consequences. in D. 7) k(D,y) ~ , f y2 do where do fixed D ~(D) is ( m - l ) - d i m e n s i o n a l surface area on y, and varying when the function subtended by D ~(D,y) v = IxlPy(x/Ixl) at the origin. 8) attains l(D) D. Then for its m i n i m u m value ~ = i8 harmonic in the cone Here = U(p+m-2).

Functions, slight i m p r o v e m e n t s harmonic p o l y n o m i a l tracts, opening to c o n s t r u c t large n u m b e r s k in the way that the of a n g u l a r We proceed a relatively to the case sharp, cones sections s quite well are no longer is the circular congruent having sharp can o b t a i n for this right FOR [3] yield ~ extends n in R m, n ~ i. 3) : (m-l)p + q, 0 ~ q < m-l. Then 2pm-l-q(p+l) q if m : 2. 5) x F r o m this COROLLARY. we can deduce We have for Z(k,m) ~(k,m) where [y] 4.

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