On the Critical Points of Functions Possessing Central Symmetry on the Sphere

On the Critical Points of Functions Possessing Central Symmetry on the SphereAuthor(s): J. L. WalshSource: American Journal of Mathematics, Vol. 70, No. 1 (Jan., 1948), pp. 11-21Published by: The Johns Hopkins University PressStable URL: http://www.jstor.org/stable/2371928 .

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ON THE CRITICAL POINTS OF FUNCTIONS POSSESSING CENTRAL SYMMETRY ON THE SPHERE.*

By J. L. WALSH.

The object of this note is briefly to consider rational and harmonic functions defined on a sphere (the Neumann sphere) and which in each pair of diametrically opposite points take the same value. We establish results concerning the location of the critical points, namely the zeros of the derivative of a rational function and the points where both first partial derivatives vanish for a harmonic function.

1. The symmetry required here is precisely the symmetry required in Klein's classical model of elliptic geometry. Theorem 1 is the analog of results already established for euclidean and hyperbolic geometries:

THEOREM 1. Let f (z) be a rational functiont of z whose zeros and whose poles occur in pair-s in diametrically opposite points of the sphere. Let P be an arbitrary point of the sphere, and let S denote the (open) hemisphere containing P whose pole is P. Suppose the great circle C through P separates all zeros of f(z) in S but not on C from all poles of f(z) in S not on C, where we assume f (z) to have at least one zero or pole interior to S not on C. Then P is not a finite zero of f'(z) unless it is a multiple zero of f (z).

Corresponding to an arbitrary rational function f (z), either on the sphere or in the extended plane, we consider a fixed particle at each zero, repelling with a force equal to the inverse distance, and a fixed particle at each pole, attracting with a force equal to the inverse distance; multiple zeros and poles are represented by multiple particles. It then follows 1 that the finite critical points of f (z) are precisely the positions of equilibrium in the field of force and the multiple zeros of f (z).

We choose, as we may do with no loss of generality, the point P as the origin: z = 0 in the Gauss plane, and the image of S as S': I z I < 1. The image of C is a line C' through P.

If z, is a zero of f(z), so is the point - 1/z2, and the forces at P due

* Received May 21, 1947. 1 B6cher, Proceedings of the American Academy of Arts and Sciences, vol. 40 (1904),

pp. 469-484; Walsh, Transactions of the American Mathematical Society, vol. 19 (1918); pp. 291-298.

11



12 J. L. WALSH.

to the corresponding particles are respectively - 1/z, and z1; the sum of these forces is

( I ) Z1 ( /Z1) - 21 ( ~~Z2

)

so that, if z1 lies interior to S', this force is directed from z1 to P. If z1 lies on the unit circle, this force is zero. Under the conditions of Theorem 1, each pair of zeros of f (z) not on the unit circle has but one member z1 interior to S'; the total corresponding force exerted at P due to the pair of particles is directed from z1 toward P, hence if z1 does not lie on C' this force has a non-vanishing component orthogonal to C'; each pair of poles of f (z) not on the unit circle has but one member Z2 interior to S', and the total corresponding force exerted at P is directed from P toward Z2, hence if Z2 does not lie on C' this force has a non-vanishing component orthogonal to C' in the same sense as the component of the force due to each pair of zeros of f(z) not on C'. Since f(z) has at least one zero or pole interior to S' but not on C0, the point P cannot be a position of equilibrium, hence cannot be a zero of f'(z) unless it is a multiple zero of f(z). Theorem 1 is established.

Theorem 1 is analogous to a result on rational functions in the euclidean plane [Bocher and Walsh, loc. cit.], and to a result on rational functions in the hyperbolic plane.2

We ilote incidentally that if we modify the hypothesis of Theorem 1 so as to allow all zeros and poles of f(z) to lie on the great circle one of whose poles is P, and if the zeros and poles of f(z) are chosen on this great circle, then P is a position of equilibrium and critical point of f(z). Likewise if the hypothesis of Theorem 1 is modified so as to permit all zeros and poles of f(z) to lie on C', and if all zeros and poles of f(z) do lie on C', then P may be a critical point of f(z) even if not a multiple zero of f(z) ; indeed P may be a point of symmetry for the zeros and poles of f(z).

We proceed to a simple application of Theorem 1. By a circular region we mean a closed region of the plane or sphere bounded by a circle. Where confusion will not result, we use the same notation for a circular region as for its boundary. Let the function f (z) possess the symmetry required in Theorem 1, let the zeros of f(z) lie in two diametrically opposite circular regions C0 and C3 of the sphere, and the poles of f (z) lie in two diametrically opposite circular regions C2 and C4 disjoint from C0 and C3. Construct the circular region Sk (k = 1, 2, 3, 4) which is the locus of points (assumed not empty) P such that every point of the region Ck is at a spherical distance from P not greater than 7r/2. Thus the circular regions Ck and Sk have the

2 Walsh, Bulletin of the American Mathematical Society, vol. 45 (1939), pp. 462-470.



CRITICAL POINTS OF FUNCTIONS POSSESSING CENTRAL SYMMETRY. 13

same poles, and the measures of their spherical radii are. complementary. Denote by T the open region (assumed not empty) common to Si and S2.

If P is a point of T, the open hemisphere whose pole is P and which contains P contains the regions C0 and C2, and contains no point of 03 or C4. Taken together with suitably chosen arcs of C0 and C2, the great circles r1 and r2

tangent to C0 and C2 and separating those circles separate T into three regions R1, R2, R, of which R1 and R2 may be empty. The region R consists of all points P in T which lie on great circles which separate C, and C2, and R is open. The region R1 closed with respect to T is disjoint from R, and consists of all points of T not in R which lie on great circles cutting both C0 and C2, but on the maximal arcs of those circles bounded by points of C, and of the boundary of T, arcs in T containing no points of C2; the region R2 is similarly defined by permuting subscripts. The regions R1 and R2 may contain the whole or parts of the regions C, and C2 respectively. Both R1 and R2 are convex with respect to great circles. It is an immediate consequence of Theorem 1 that all finite critical points of f (z) in T lie in R1 and R2; no finite critical point of f (z) lies in R. We have constructed the geometric configurations C1, C2 R1, Ri2, R, T for the purpose of simplicity of exposition; under suitable conditions corresponding regions Ri, R2, R, T may be used without the introduction of circular regions Ck.

Another result is also riot difficult to establish:

THEOREM 2. Let f (z) be a rational function of z whose zeros and whose poles occur in pairs in diametrically opposite points of the sphere. Let P be an arbitrary point of the sphere, and let S (considered closed) denote the hemisphere containing P whose pole is P. Let all zeros [or poles] of f (z) in S lie exterior to the circular region CO containing P and bounded by a small circle in S of the sphere whose pole is P. Let all poles [or zeros] of f (z) in S lie in the circular region C2 interior to CO but not containing P. Then P is not a finite critical point of f (z).

We prove Theorem 2 by interpreting the configuration in the plane instead of on the sphere with the image of P (also denoted by P) in z =- O, the image of S as S': I z _ , the image of C0 as C'1: I z I ?pl, and the image of 02 as C'2. For definiteness suppose the zeros of f(z) in S to lie exterior to CO.

Let 2n denote the degree of f (z). For each zero z1 of f (z) interior to S' it follows from (1) that the force at P for the corresponding pair of zeros of f (z) is in magnitude less than

- pip P1



14 J. L. WALSH.

so that the total. force at P due to particles at all the zeros of f (z) is less than n( 1 -p12)/pi.

Construct an auxiliary circle CO containing in its interior the interior of C'2, so that Co passes through P and lies interior to C'l; we assume for definiteness the center of Co to be the point p/2, on the positive half of the axis of reals. If a pole (r, 0) of f(z) lies in or on Co (r # 0), the force exerted at P by the corresponding pair of particles is in magnitude (1/r) -r, and the component of this force in the positive horizontal direction is (1 - r2) (cos 0)/r. But since the pole lies in or on CO we have

1 ? cos 0 >r r>r(l-p) (1 -r2)cos 0 > 1 p p(1 -r2)' r = P

this inequality merely expresses the fact that for each pair of poles, of which one representative lies in or on CO (but not at P), the corresponding force exerted at P has a horizontal component in the positive sense not less than it would be if the representative coincided with the point (p, 0). Thus under the conditions of Theorem 2 the total force at P due to the particles at the 2n poles has a horizontal component in the positive sense which is greater than the total force at P due to the particles situated at the 2n zeros of f (z). Thus P cannot be a position of equilibrium nor a multiple zero of f (z), so that P is not a critical point of f(z).

We mention an immediate application of Theorem 2. Let f (z) have the symmetry required in Theorem 2, and let all zeros [or poles] of f (z) lie in diametrically opposite circular regions C0 and C3 of the sphere. Let no pole [or zero] of f(z) lie in the circular region C2 which has the same poles as C' and which contains C1. Denote by pi and P2 (< 7r/2) the spherical radii of

CG and C2, and suppose p2 > 3pi. Then the (open) zone bounded by Ci and the circle C in C2 having the same poles as C, and spherical radius (p2 - pi)/2

contains no finite critical points of f (z). If P is an arbitrary point of this zone, whose spherical distance from the pole of the region C0 is denoted by p, the circular region containing P whose pole is P and spherical radius p + pi

contains C, but contains no point of the circle C2; the hemisphere containing P whose pole is P contains no point of 3,; the conclusion follows from Theorem 2.

In the proof of Theorem 2 we notice that if r is the great circle through P orthogonal to the great circle through P orthogonal to C2, then the zeros of f (z) in S which are on r or separated from C2 by r need not lie exterior to

Cj; the proof already given is valid if we notice that pairs of zeros represented by points in S separated from C2 by r yield a non-negative component at P in the direction and sense from P toward the center of C2. Thus we have the




COROLLARY. Let f (z) have the symmetry required in Theorem 2, let P be an arbitrary point of the sphere, and let S be the closed hemisphere containing P whose pole is P. Let each zero [or pole] of f(z) in S lie either exterior to the circular region C, containing P and bounded by a small circle of the sphere in S whose pole is P, or lie not at P but on a great circle r through P, or be separated by r from a circular region C2 in S not containing P whose boundary is orthogonal to the great circle through P orthogonal to r. Let all poles [or zeros] of f (z) in S lie in C2. Then P is not a finite critical point of f (z).

As a simple application of Theorems 1 and 2 we establish

THEOREM 3. Let f (z) be a rational function of z whose zeros and whose poles occur in pairs in diametrically opposite points of the sphere. Let all the zeros lie in diametrically opposite circular regions Cl and C3 and all the poles lie in diametrically opposite circular regions C2 and C4, where all the regions Ck are mutually disjoint. Denote by Z1 and Z2 the (closed) zones of the sphere which are the loci of the great circles whose poles lie in C] and C3

and in C2 and C4 respectively. Suppose that for the circles Cl and C2 (and likewise C2 and C3), the (spherical) length of the common tangent T (chosen as the shorter arc of a great circle separating the regions C0 and C2) is not less than the sum of the (spherical) diameters of those circles.3 Then all finite zeros of f'(z) lie in the regions Ck and in Z1 and Z2.

Let a point P of the sphere not in a region Ck or a zone Z7k be a finite zero of f' (z) ; we shall reach a contradiction. The hemisphere S whose pole is P and which contains P must contain in its interior the whole of one of the regions C0 and 03, and no point of the other of those regions, for P does not lie in Z1; similarly for the regions C2 and C4. Let us suppose S to contain C0 and C2.

No great circle through P can separate C] and C2, by Theorem 1. For

definiteness suppose the pole of C] in S to be nearer P than that of C2 in S. There exists a great circle C through P which cuts C0 and C2 at supplementary angles, those circles being oriented in the same sense on the sphere; thus C may be defined as the great circle through P and through the intersection of the common tangents T to C] and C2. Then C cuts C] and C2 in such a way that the points P, A1, B1, A2, B2 lie in that order on an arc of C in S, where

Ak and Bk lie on Ck; the points Ak and Bk do not coincide unless C is tangent

3 This condition is merely a convenient one for use in the proof; it may be replaced by other conditions less restrictive.



16 J. L. WALSH.

to Ck at A*. The spherical (listance A1B2 is not less than the length of T, which by hypothesis is not less than the sum of the diameters of C6 and C2.

The circle r whose pole is P and whose radius is PB2 less the spherical measure

of the diameter of C2 cannot cut C2. Also, P cuts C between A1 and B2 at a point whose distance from A1 is not less than the diameter of Cl, so that r cannot cut Cl, and C0 is separated by r from C2. It follows from Theorem 2. that P is not a critical point of f (z) ; this contradiction completes the proof of Theorem 3.

2. Theorems 1 and 2 are both results which, referring to the zeros and

poles of f (z) located in a hemisphere S which contains its pole P, assert that if the zeros of f (z) lie in a suitable circular region CL and if the poles of f (z) lie in a circular region C2 which has no point in common with Ci

and satisfies auxiliary conditions, then P is not a critical point of f (z). The question suggests itself whether some auxiliary conditions are here necessary, a question which we answer in the affirmative:

THEOREM 4. There exists a rational function f(z) of degree fotr whose

zeros and whose poles occur in pairs in diametrically opposite points of the

sphere; there exists a point P which is a finite critical point of f (z) ; there exist two diametrically opposite circular regions C0 and C2 each of which contains precisely two zeros of f (z) and which do not contain P nor intersect the great circle r one of whose poles is P; the poles of f (z) are double and lie exterior to C0 and C2.

Theorem 4 is trivial if we omit the requirement that C0 and C2 shall not intersect r, for if P is arbitrary and all zeros and poles of f (z) lie on r, then P is a position of equilibrium and hence a critical point of f (z).

Let P be given, together with the arbitrary diametrically opposite circular

regions C0 and C2 which neither contain P nor intersect r. We fasten our

attention on the representatives in C0 of the pairs of zeros of f(z). Given two zeros z1 and Z2 of f (z) in Cl, the corresponding two pairs of zeros of f (z) can be replaced by a single pair of double zeros (possessing the required symmetry) without altering the corresponding force exerted at P. If P is the point z 0 and r is the unit circle, the equation to be solved for one of these

double zeros zo is

(2) (z1 - 1/Z1) + (Z2 - 1/2)- 2 (zo -/O).

It will be noticed that equation (2) with the first member not zero defines

zo in I zO I < 1 uniquely; we merely write



CRITICAL POINTS OF FUNCTIONS POSSESSING CENTRAL SYMIMETRY. 17

zO (I z?) A =,= 0, arg zo argA,

z (1 O0 ) =AA; _ZoZo2

this last equation defines I zO I uniquely, subject to the condition I zo < 1. If for suitable choice of C,, z1, and Z2, the point zo cannot be chosen in the region C,, we need merely choose the double poles of f(z) in zo and in -1/4 in order to establish Theorem 4.

The entire question, now reduced to the location of the point z0,, can be further transformed. To every point z1 of the circular region C' we make correspond the point w1 z - (1/z,); if the locus of the points w1 is convex, then to every pair of points z1 and Z2 in C' corresponds by (2) a point zo also in C, equivalent to z1 and Z2 in the sense that a double particle at zo (together with its mate at - 1/zo) exerts the same force at P as do the pairs corresponding to z1 and Z2. On the other hand, if the locus of the points w1 is not convex, then for suitably chosen z1 and Z2 in C, the pair zo and -1/4 defined by (2) has no representative in Cl, and the double poles of f(z) in Theorem 4 may be chosen in the points zo and - 1/z4 exterior to C', and Theorem 4 is established. We proceed, then, to study the convexity of the locus of w1. Here it is a slight convenience algebraically to choose C, as exterior to I z = 1; this is a choice merely of studying one of the two members of a symmetric pair rather than the other.

If the point z is exterior to I z = 1, the point w defined by the equation w = z o (1/2) = z(zz - 1)/zz lies on the half line from the origin through z, and (it is sufficient here to investigate real z) I w I increases as I z I increases. Let C, be the circle I z - a r with a > 1 + r, since C, is exterior to the unit circle. It is no loss of generality to choose a real, which we do, setting z = a + re0, w - u + iv. Straightforward algebraic computation of du/dcp and dv/d4 then yields

dv -(a2+ r2) cos -2ar + cos p(a2 + r2 + 2ar cos p)2 du sin 4 (a2 r2) + (a2 + r2 + 2ar cos k)2]

This denominator vanishes only when sin 4 0, for we have a > r. Further computation shows that the algebraic sign of d (dv/du) /dc is the same as the algebraic sign of the function

F(p) (a2 + r2 + 2ar cos ) 3-2r(r + a cos p) (a2 + r2 + 2ar cos 4)

+ 8a2r sin2 p(r + a cos o) - (a2 - r2);

in writing this equation we have suppressed a factor a2 + r2 + 2ar cos4,

2



18 J. L. WALSH.

which is equal to zz and positive. It may be verified that F(0) and F (7r) are

both positive, and that when r is small in comparison with a the function F(Q) is positive for all values of 4.4

We now choose specifically r 1, cos 4 2/a, and set a = 2 + E,

E > 0. Then we have F(q) = - 16E + , where only powers of E higher

than the first are omitted. Consequently for suitably chosen E we have

F(qp) < 0, the locus of w is not convex, and Theorem 4 is established.

3. Theorems 1, 2, and 3 have application to the studv of harmonic

functions defined on the surface of the sphere and possessing there suitable

symmetry. The analog of Theorems 1 and 2 is

THEOREM 5. Let R be a region of the sphere bounded by a finite number

of mutually disjoint Jordan curves, and let R possess central symmetry. Let

the function U(x, y) be harmonic in R, continuous in the corresponding

closed region, equal to zero on a symmetric set JO and to unity on the remaining

set J1 of the curves bouinding R. Let P be an arbitrary point of the sphere, and let S denote the closed hemisphere containing P whose pole is P.

Suppose that the great circle C through P separates all points of JO in S

from all points of J1 in S. Thenl P is not a critical point of U (x, y).

Suppose that all points of JO [or of J1] in S lie exterior to the circular

region C' containing P and bounded by a small circle in S of the sphere whose

pole is P. Suppose that all points of J1 [or of JO] in S lie in the circular

region C2 interior to C0 butt not containing P. Then P is not a critical point of U(x,y).

The functioln U (x, y) is the harmolnic measure of J, in the point (x, y)

with respect to the region R, and the function 1 - U (x, y), which has the

4Iiideed, F (0) is positive whelnever r + a cos 0 is positive. Thus for suitably restricted circles CO and also for suitably chosen other regions R1 whose boundaries may contain arcs of circles Cl, the locus of w is convex. Under such conditions let us say that R has Property a; this property here depends on a particular point P. If the rational function f (z) has the requisite symmetry, if P is an arbitrary point, if S denotes the closed hemisphere containing P whose pole is P, and if disjoint closed regions R1 and R2 interior to S not containing P and having property a contain respectively all zeros and all poles of f (z) in S, theni P is not a critical point of f (z); compare Theorem 3. By way of proof it is sufficient to note that in the field of force, without altering the total force exerted at P, all positive particles in S may be concentrated at a single point which lies in R, and all negative particles in S nmay be concentrated at a single point which lies in R2, without altering the symmetry. 'Thiis the total force at P cannot vanish, and P cannot be a critical point of f (z).




same critical points as U (x, y), is the harmonic measure of JO in the point (x, y) with respect to R.

We represent U (x, y) by integrals taken over the sets of curves JO and J1 if these curves are analytic, and otherwise taken over neighboring sets of analytic level curves of U (x, y), curves J'o and J'1 in R found from JO and J1 by slight deformations: 5

U (X, y) -K =-(1/2ir) f log r(0U/&v)ds- 1/27r) flog r(OU/0v)ds,

(x,y) in R;

here v indicates exterior normal, and K is a suitable constant. If we introduce the new variable ar by setting

da- (OU/0v)ds on J'o, du-= (OU/0v)ds on J'1, T- =fdu= f do,

we may write

U (x, y) -xK 2 log rct - log rd

T lim log r, n + log r2 + *+ log rn,, 27r n--o n

--lim lim pni + log pn2 + + log pnn 27 n--o n

where rnk = z ank I = z -/fnk I , and ank and ftnak are suitably chosen

on Jo' and J'1 respectively. Convergence is uniform on any closed preassigned set interior to R.

By virtue of the symmetry of R and U (x, y), the sets of curves J'o and J'i

may also be chosen to be symmetric in the center of the sphere, and likewise

the sets of points ank and Bnk, where n is taken as even. In the neighborhood of an arbitrary point of R, a constant multiple of the function U (x, y) - K

is thus the uniform limit of the logarithm of a sequence of rational functions Rn (z) whose zeros and poles lie outside of that neighborhood; indeed the functions Rn (z) possess the symmetry demanded of U (x, y), and their zeros

and poles can be chosen to lie respectively in the regions required in Theorem 5 for Jo and J1. Theorems 1 and 2 apply to the functions Rn (z). If a point P satisfies the conditions of Theorem 5, so do all points in a suitable neighborhood of P; no point of such a neighborhood is a critical point of an Rn (z). Each critical point of U(x, y) in R is a limit of critical points of the Rn,(z), so that Theorem 5 follows.

r Walsh, Interpolation and Approximation (New York, 1935), ? 8. 7.



20 J. L. WALSH.

The reader will have no difficulty in formulating and proving the analog of Theorem 3.

4. The relation between Theorem 1 and its analog in hyperbolic geometry is closer than mere analogy. Theorem 1 deals with the point P: z = 0 as a possible critical point of the rational function f (z), whose zeros are the points ak and- l/k (k 1, 2, . . , n), and whose poles are the points Ok and

-1/,k (Ic = 1, 2, , n); we choose this notation so that we have ?ak 1 < j I I < 1. So far as concerns the force exerted at P, the field of

force corresponding to f(z) is equivalent to the field of force corresponding to the rational function F(z) whose zeros are the points ak and 1//3k, and whose poles are the points /k and 1/k (k = 1, 2, , n); we omit from this enumeration points ak, /3k, l/lk, and 1/17 of modulus unity. The zeros and poles of F(z) interior to the unit circle C are precisely those of f (z) interior to C; the zeros and poles of F(z), exterior to C are precisely the negatives of the poles and zeros respectively of f(z) exterior to C. It will be noted that the zeros of F(z) are the inverses of the poles of F(z) with respect to C; the function F(z) is of constant modulus on C. The point P is a critical point of f (z), whether a multiple zero or a position of equilibrium in the field of force, when and only when P is a- critical point of F(z). It is thus a consequence of Theorem 1 that if a line L separates the zeros of F(z) interior to C: z 1 not on L from the poles of F(z) interior to C not on L, and if C contains at least one zero or pole not on L, then P is not a critical. point of F(z) unless P is a multiple zero of F(z). The property possessed by F(z), that its zeros are inverse with respect to its poles in a circle, is characteristic of all rational functions having constant modulus on a circle, and is invariant under linear transformation. We thus have from Theorem 1 by proceeding from a given F (z) to the function f (z), a result on a rational function F(z) of constant modulus on an arbitrary circle, a result involving the images of lines L through P and the analog (loc. cit.) of Theorem 1 for hyperbolic geometry:

If the rational function F(z) has its poles inverse to its zeros in a circle C, if P is an arbitrary point interior to C, and if a non-euclidean line L through P separates all the zeros of F(z) interior to C not on L .from all the poles of F(z) interior to C not on L, where we suppose at least one such zero or pole to exist, then P is not a critical point of F(z) unless P is a multiple zero of F(z).

At first sight it might appear as if this last result were established only for a rational function F(z) of special type, having the same number of zeros




as poles interior to C. However, in proceeding from a given F(z) to the corresponding f(z), we may provide the latter artificially with any number of zeros or poles situated on C itself and possessing the required symmetry. With this interpretation, these two analogous theorems of the elliptic and hyperbolic geometries respectively can be considered equivalent, in the sense that either can be trivially proved from the other.

The result on hyperbolic geometry is of some interest in the theory of functions, for by a suitable conformal map it applies to the study of a function meromorphic in an arbitrary simply connected regioni, of constant modulus on the boundary.

Results of the present note other than Theorem 1 can also be applied by the reader in the study of hyperbolic geometry. We state merely the analog (the equivalent) of Theorem 2: Let R be a simply connected region provided with a non-euclidean hyperbolic geometry by means of a conformal map onto a circle. Let P be a point of R, let Ro be the closed subregion of R containing P bounded by a (non-euclidean) circle whose (non-euclidean) center is P, and let R1 be a closed subregion of Ro not containing P bounded by a (non- euclidean) circle. If f(z) is meromorphic in R and of constant modulus on the boundary of R, if the zeros [or poles] of f (z) in R are n in number and lie exterior to Ro, and if the poles [or zeros] of f(z) in R are not fewer than n in nuumber and lie interior to R1, then P is not a finite critical point of f (z).

HARVARD UNIVERSITY.



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On the Critical Points of Functions Possessing Central Symmetry on the Sphere