Properties of the Newtonian potential

This extremely technical article treating in detail of the properties of a potential inversely proportional to the distance is addressed above all to the students and to the researchers in physical geodesy and geophysics, but can also interest the physicists theorists. Its reading requires a turn of mind directed towards mathematics applied.

Potential of gravitation and force of gravitation

One calls the potential gravific of a mass distributed in volume B of a body (Earth in our case) a Newtonian potential . This potential plays a so important part in physical geodesy which it seems necessary to establish in detail the properties of them. For this purpose we suppose that the density ρ body attracting B is in general a continuous function, but that it can exist a finished number of surfaces in B on which the density is discontinuous. Such a function is called continuous per pieces . We will not make distinction here between specific space and the associated vector space, and we will write for the unspecified points P (x₁, x₂, x₃) and Q (y₁, y₂, y₃) also X and there , respectively. We will in general indicate the elements of volume by dτ and the elements of mass by DM. We thus have the relation: DM = ρ dτ.

Consequently, while paying our attention on the point Q in particular, we will indistinctly use dτ (Q), D ³ there or dy₁dy₂dy₃ for the element of volume in Q, and DM (Q) or DM ( there ) for the element of mass in Q. In vectorial notation we will thus write

V (P) = V ( X ) = G ∫_B ρ ( there ) | there - X |^-1 D ³ there

for the potential of gravitation and

X_i (P) = X_i ( X ) = G ∫_B ρ ( there ) (y_i-x_i) | there - X |^-3 D ³ there

for the IE component of the force of gravitation. If we pose ρ (Q) = 0 in any point Q external with the body B, i.e. in empty space, we can extend integrations in the two preceding expressions to any regular field D containing B. Any point pertaining to the field of integration is called a not interior (or not interns), all the other points are called external points (or external points).

That is to say R = there - X the vector uniting P with Q. We have then

r_i = y_i-x_i for i=1,2,3.

The distance D (P, Q) between P and Q can be written any in the following ways

D (P, Q) = | there - X | = ^1/2 = R = (r₁ ² +r₂ ² +r₃ ²) ^1/2 = (r_kr_k) ^1/2,

by implying implicitly that in the last term the summation is carried out for all the values of the dumb index K which is repeated (convention of summation of Einstein). It is often advantageous to write the integrals higher in the form of Intégrale of Stieltjes, namely:

V (P) = G ∫_M r^-1 DM,

X_i (P) = G ∫_M r_i r^-3 DM, where M is the total mass of the body B.

Existence and continuity of the Newtonian potential and its derivative in space

We will show now that the potential V (P) and its derivative first exist and are continuous in all space for continuous distributions of density per pieces. As the number of surfaces of discontinuity is finished, the body B can be subdivided in a number finished NR of smaller bodies B_k of M_k masses such as

M = ∑_k=1→N M_k,

in each one of which the density is continuous. One can thus write

V (P) = G ∫_M r^-1 DM = G ∑_k=1→N ∫_{M_k} r^-1 DM.

in each one of which the density is continuous.

Existence and continuity of the Newtonian potential and gravity outside the matter

If the point potentié P is outside B, each integral ∫_{B_k} ρ ( there ) | there - X |^-1 D ³ of y' exists, because the function ρ ( there ) | there - X |^-1 is continuous in the area of integration. Since each integral exists separately, their nap also exists, and consequently V exists at the external points. Moreover, since the intégrand in the expression of V (P) is positive everywhere of the same G, there is GM/r_min ≤ V (P) ≤ GM/r_max. Here, r_min and r_max indicate respectively the distances minimum and maximum between the attracting body and the point attracted P external with the body. When P moves away from B, at the same time r_min and r_max grow indefinitely. One thus sees that V has limit zero ad infinitum, whatever the direction considered. In more precise terms, one says that |V| behaves regularly ad infinitum and decrease like d^-1 (O, P). Indeed, is Q an unspecified interior point. Then the distance D (O, Q) is limited and |D (O, P) - D (P, Q)| ≤ D (O, Q), so that

D (O, P) V (P) = G ∫_B ρ (Q) {1 - /d (O, P)}^-1 dτ (Q) = G ∫_B ρ (Q) dτ (Q) + terms of order 1/d (O, P).

By completely similar arguments it is possible to prove that X_i exists in space external with the body B and that intensity of the field of gravity

X = (X₁ ² +X₂ ² +X₃ ²) ^1/2

vary at long distance like d^-2 (O, P). Moreover, it is obvious that the potential and the intensity of gravity are continuous in external points. By the definition-even of a Derived , which in the theory of the potential is generally considered like a directional Dérivée, derivation under the sign integral is entirely justified outside the material body. Are H the length of a small segment of right-hand side and X _i^o an unit vector carried by the axis Ox_i. Then, by definition:

∂_iV = G lim_h=0 h^-1 ∫_M dM/r (''' X ''', ''' there '''),

thus

∂_iV = G lim_h=0 ∫_M {∂_i ½ H ∂_i ² [1/r (''' X ''' +ϑh ''' X ''' _i^o, ''' there ''')} DM ( there ),

where ϑ is a certain number ranging between 0 and 1. Like

∂_ir^-1 = (y_i-x_i) r^-3,
∂_i ² r^-1 = 3 (y_i-x_i) ² r^-5 - r^-3,

one has

|∂_i ² r^-1 ( X +ϑh' x' _i^o, there )| < 4 L^-3

if L is the distance between the element of mass DM ( there ) and the P' point positioned in X +ϑh' x' _i^o. Let us suppose that Q' is the point nearest to P' such as Q' still forms part of B. Then D (P', Q') is the smallest possible value of L, and

| ½ H ∫_M ∂_ir^-1 ( X +ϑh' x' _i^o, there ) DM ( there ) | < ∫_M 2 |H| DM ( there ) d^-3 (P', Q') = 2 M |H| d^-3 (P', Q').

When H tends towards zero, the P' point is driven towards P. For this reason, the distance minimum D (P', Q') of P' to the body can change. However, as P is outside the area B, there exists a value minimum of D (P', Q') nonnull when |H| is sufficiently small. That is to say d_min lim_h=0 | ½ H ∫_M ∂_ir^-1 ( X +ϑh' x' _i^o, there ) DM ( there ) | < 2 M |H| d_min^-3

cancel yourself with h. We can thus conclude that in any point external the gradient of the potential of gravity exists and, because of the foregoing expressions providing X_i and ∂_ir^-1, represents gravity well:

∂_iV = G ∫_M ∂_ir^-1 DM = X_i.

Equation of Laplace

In fact, one can extend this demonstration successively to derivative of higher natures and thus to prove that in an area deprived of matter (i.e. in space empties ), the potential gravific has derivative partial continuous of all the kinds and is analytical.

In particular, if we consider the derivative partial of order 2, we find thanks to the foregoing relation providing ∂_i ² r^-1 that

(∂₁ ² + ∂₂ ² + ∂₃ ²) r^-1 = 3 r ^-5 - 3 r ^-3 = 0.

We end thus to famous the equation of Laplace

(∂₁ ² + ∂₂ ² + ∂₃ ²) V = G ∫_M (∂₁ ² + ∂₂ ² + ∂₃ ²) r^-1 DM = 0.

This one is an partial derivative equation of the second order which must be satisfied by the potential gravific in the areas with space deprived of matter. She plays a fundamental role, not only in geodesy, but in practically all sciences which admit a mathematical description of the phenomena. Pointing out the to us physical significance of the operator Laplacian (∂₁ ² + ∂₂ ² + ∂₃ ²), we know that the functions which obey the equation of Laplace, such as the potential gravific in the vacuum, have the property to be average functions in the sense that the value of such a function in a point is the average of the values of this function in a sufficiently restricted vicinity of this point. Such a function is still called harmonic function .

Existence of the Newtonian potential and gravity inside the matter

To establish the properties of the potential gravific inside a material body, in particular inside the Earth, the demonstrations become more complicated. That is due to the fact that for interior points, the integral V ( X ) = G ∫_B ρ ( there ) | there - X |^-1 D ³ is unsuitable there following the singularity which occurs when | there - X | = 0. Consequently, derivation under the sign of integration is not allowed any more without justification the interior points. It is the need for such a justification which complicates the demonstrations. Indeed, even the existence and the continuity of the potential in points where there exists matter are not any more of the obvious properties, but require a proof. Thus let us suppose that P is an interior point. In order to simplify the demonstrations we will pass, without loss of general information, of the Cartesian frame of reference under consideration until now at a spherical frame of reference (R, θ, λ) admitting P like origin. One obtains as follows:

r₁ = y₁-x₁ = R sin θ cos λ, r₂ = y₂-x₂ = R sin θ sin λ, r₃ = y₃-x₃ = R cos θ.

The vector X being fixed, the element of volume in Q in terms of the variables R, θ, λ is D ³ there = D ³ R = R ² sin θ Dr. dθ dλ. Having defined the density as being null in-outside B, we can write

V ( X ) = G ∫_D ρ ( X + R ) r^-1d ³ R ,

where the field of integration D has a spherical border of ray L, centered on the point P. We take for L the longest distance between two arbitrary points belonging to the material body of finished size B. Either ρ_max the maximum value of |ρ| in D. Alors

V (P) ≤ 2πG ρ _max L ².

Thus, the unsuitable integral V ( X ) above converges uniformly compared to the parameters x₁, x₂, x₃, which implies that the potential V exists in each point. In a completely similar way, we can prove that unsuitable integrals defining the components X_i (P) (i=1,2,3) of gravity exist. Indeed, by still regarding spherical coordinates and by keeping the same notations as above, we find

|X_i (P)| = G | ∫_D r_i ρ ( X + R ) r^-3d ³ R | ≤ 4πG ρ _max L.

This proves uniform convergence, and thus the existence, of the unsuitable integrals X_i.

Continuity of the Newtonian potential and gravity inside the matter

We show then that the potential V is continuous in a P_o point of B. It is not a restriction essential only to suppose that P_o is inside B. Indeed, we saw that we can increase B by posing ρ (Q) = 0 in the area which one adds. The traditional reasoning then consists in breaking up the field B into two under-fields, namely B-b and B. The under-area B is small swell centered on P_o. We write

V (P) = V₁ (P) + V₂ (P)

with the help of

V₁ (P) = G ∫_b ρ (Q) d^-1 (P, Q) dτ,
V₂ (P) = G ∫_B-b ρ (Q) d^-1 (P, Q) dτ.

However, for ε > 0 given arbirairement small, we can take B sufficiently small so that

|V₁ (P)| < ⅓ ε,

independently of the position of P. With this intention, it is enough to take the ray r_b B smaller than ^1/2, since in this case it comes

|V₁ (P)| < 2πG ρ _max r_b ².

It follows that for such a sphere B one has

| V₁ (P) - V₂ (P_o) | < ⅔ ε.

Then, with B fixed, there exists a vicinity of P_o such as, if P belongs to this vicinity and Q belongs to B-b under-volume, we have

| d^-1 (P, Q) - d^-1 (P_o, Q) | < ε ^-1.

Thus, when P is in this vicinity, we find

By combining the inequalities for V₁ and V₂, we have

| V (P) - V (P_o) | < ε.

Thus V is continuous in P_o, and consequently in all space.

As previously, one will establish the continuity of gravity by a similar partition of B in two under-areas. Indeed, that is to say

X_i (P) = X_i⁽¹⁾ (P) + X_i⁽²⁾ (P),

with

X_i^k (P) = G ∫_{b_k} ρ (Q) e_i d^-2 (P, Q) dτ (Q), K = 1,2.

Here we use the following notations: b₁ = B, b₂ = B-b, E is the unit vector PQ ⁰:

|e_i| ≤ (e₁ ² +e₂ ² +e₃ ²) ^1/2 = 1.

For all ε > 0 fixed arbitrarily small, we can take the ray of B smaller than ε ^-1. Then

| X_i⁽¹⁾ (P) | < ⅓ ε

independently of the position of P, and

| X_i⁽¹⁾ (P) - X_i⁽¹⁾ (P_o) | < ⅔ ε.

Moreover, with B fixed but sufficiently small and containing the points P and P_o, and with Q pertaining to volume B-b, we have

| d^-2 (P, Q) - d^-2 (P_o, Q) | < ε'/

and

By combining the inequalities for X_i⁽¹⁾ and X_i⁽²⁾, we establish the continuity of the field of gravific force in any interior point P_o and, consequently, the continuity of gravity in all the space filled or not with matter.

Existence and continuity of the derivative partial of the potential inside the matter

However, it is not obvious without complementary investigations that the components of X_i force are equal to the derivative partial ∂_iV (I = 1,2,3) at the points where there is matter. Indeed, the usual conditions making it possible to derive under the sign ∫ are not met by the unsuitable integrals. Nevertheless, the X_i relation = ∂_iV remains in this case. To prove it, let us consider two points P and P_o interiors with B, occupying compared to the origin O the positions X and X + H X _i^o, respectively. That is to say Q an attracting point occupying the position there . Compared to P, Q is located in R = there - X . Let us consider now the expression

I (H) = X_i (P) - h^-1 = G ∫_B ρ ( R ) D ³ R ,

where

r_i = y_i - x_i,
R = D (P, Q) = | there - X | = (r₁ ² +r₂ ² +r₃ ²) ^1/2,
r_o = D (P_o, Q) = | there - X - H X º_i = (R ² +h ² - 2hr_i) ^1/2.

If we manage to show that the limit for H tending towards zero of |I (H)| than a positive number ε is smaller arbitrarily small fixed in advance, then we can conclude that the derivative partial ∂_iV of V exists and is continuous at the interior points, just like for the external points, and that its value is X_i. Thus, let us take |H| enough small so that P_o is inside the spheroidal field B of ray r_b, centered on P. the value r_b itself is selected rather small so that the ball B is contained entirely in volume B. Thus, like P and P_o is external with volume B-b, it results from the properties of the potential in external points that, whatever the value of the ray r_b, the distance D (P, P_o) = |H| < r_b can be taken sufficiently small so that

I₂ (H) = | G ∫_B-b ρ ( there ) D ³ there < ½ ε.

With regard to the contribution I₁ (H) of the integral taken on spherical volume B, it is appropriate to notice first of all that

|r_i| ≤ R

and that

r^-1 - r_o^-1 = (r_o - R) (R r_o) ^-1.

Relations expressing the triangular inequalities applied to the PP_oQ triangle it is deduced that

|r_o - R| ≤ |H|.

In addition, like

(r^-1 - r_o^-1) ² = r^-2 + r_o^-2 - 2 r ^-1 r_o^-1 > 0,

it comes

(R r_o) ^-1 < ½ (r^-2 + r_o^-2) < r^-2 + r_o^-2.

Thus, one leads to increase

| I₁ (H) | < G ∫_b (2 r ^-2 + r_o^-2) ρ ( there ) D ³ there .

While posing like habit ρ (Q) = 0 for any point Q being in-outside volume B, we can write

| I₁ (H) | < 2G ∫_b ρ r^-2 dτ + G ∫_{b_o} ρ r_o^-2 dτ,

where

b_o is a spheroidal field of ray 2r _b centered on P_o. Thus, we have

| I₁ (H) | ≤ 16πGρ_maxr_b

For sufficiently small values of r_b, i.e. for

r_b < ε/

this expression is numerically lower than ½ ε, and thus

| I (H) | ≤ | I₁ (H) | + | I₂ (H) | < ε.

In this way, we proved that the first order derivative partial of the potential exist and represent the components of gravity. This proposal is completely general. It is valid when the attracted point is outside the body which attracts, but it is also valid when the attracted point is inside this body. Another manner of being expressed consists in saying that the derivative of V first order are obtained by derivation under the sign of integration.

Existence of the derivative partial seconds of the Newtonian potential inside the matter

The situation is quite different derivative of the second order inside the body. The simple fact of supposing the density continues and limited is not enough any more to guarantee the existence of these derivative. That is seen clearly if we derive formally the expression

∂_iV ( X ) = - G ∫_B r r_i ρ ( there ) D ³ there

compared to x_j under the sign of integration, and let us try to show the existence of ∂_j∂_iV like before for V and ∂_iV, by increasing the field B in a spheroidal field D of ray L centered on P ( X ) and containing B completely. Indeed, we find then formally

∂_j∂_iV ( X ) = - G ∫_B ∂_j (r r_i) ρ ( there ) D ³ there = - G ∫_B (3 r r_i r_j - r^-3 δ_ij) ρ ( there ) D ³ there

and

|∂_j∂_iV ( X )| < ∫_D 4G |ρ (R, θ, λ)| r^-3 R ² Dr. dθ dλ.

Condition of Hölder

Here we cannot simply replace |ρ| by its upper limit ρ_max to be able to affirm than the integral of the member of right-hand side exists, because the unsuitable integral ∫_0→L r^-1 Dr. diverges. We impose for this reason on the density ρ (Q) a condition out of P which was initially introduced in 1882 by Otto Hölder, namely

|ρ (Q) - ρ (P)| ≤ has r^α
for all the points Q such as R = D (P, Q) ≤ C
where has, α, C are positive constants.

One can show that a condition of Hölder is stronger than the condition of continuity, but weaker than the condition of derivability if α < 1. A fortiori it is thus weaker than the condition of analyticity. If there exists an area B in which ρ (Q) obeys in a condition of Hölder in each point, with the same values of has, α and C, then the function ρ (Q) is known as to uniformly fill a condition of Hölder in B.

An obvious case for which a uniform condition of Hölder bracket is that of a field B \ in which density is constant, that is to say ρ (Q) = ρ_o. Let us suppose that B is a spherical volume of ray r_b and show that in this example the derivative second of V exist indeed in an interior point P. Without restricting the general information of the demonstration, we can suppose that the center of B coincides with the origin O of the axes of coordinates, and we take the vector X = COp along the axis Ox₃. In terms of the spherical coordinates (there, θ, λ), where there = (y_k y_k) ^1/2, θ = arcos (x_k y_k/xy), with X = (x_k x_k) ^1/2, the potential out of P generated by all the masses in B is provided by

V (P) = 2πG ρ _o (⅓ X ² - r_b ²).

Being a scalar function, the potential gravific is invariant compared to the changes of the frame of reference. Consequently, this formula is general. It follows that in each point inside a homogeneous spherical mass, we have

V = 2πG ρ _o (⅓ x_k x_k - r_b ²),

∂_iV = 4πG ρ _o x_i/3,

∂_j∂_iV = 4πG ρ _o δ_ji/3.

We note, in particular, that all the six derivative partial of order 2 of V exist and are continuous . Moreover, we find that

∂_i∂_iV = 4πG ρ _o.

We can now study the derivative partial of the second order of V in an interior point unspecified P of the field finished B, for a distribution of general density satisfying a condition of Hölder of the type above. As previously, is B a spheroidal field of ray r_b centered on P. We suppose that volume B is entirely contained in volume B. then Consider separately the potential V₁ (P) created out of P by all the masses in B, and the potential V₂ (P) created out of P by all the remaining masses. Since P is a point external for the evaluation of V₂, this potential has derivative continuous of all kinds out of P and is harmonic there, i.e. V₂ is a solution of the equation of Laplace in P. Ainsi, the problem is tiny room to a study of V₁.

If we write

ρ (Q) = ρ (P) +,

we see that the potential of a sphere having a continuous density out of P is the sum of two potentials: potential created by the sphere if it were filled with matter of constant density equal to that of the point P, and the potential of a sphere of which the density is cancelled in P. As we have just shown that the potential of a uniform sphere has derivative continuous of the second order, it remains us to discuss the case in which the density is cancelled out of P and meets a condition there of Hölder. By admitting that the ray r_b ball B is smaller than a certain given positive constant C, that means that |ρ₁| ≤ has r^α with R = D (P, Q) ≤ r_b ≤ C,

where ρ₁ indicates the function ρ (Q) - ρ (P). This one generates in volume B the V₁₁ potential and the field of force X_1j⁽¹⁾, J = 1,2,3. Let us define the quantity

J_ij (P) = - G ∫_b ∂_i ρ₁ ∂_j r^-1 D ³ R ,

who becomes successively

J_ij = - G ∫_b ρ₁ ∂_i (r^-3 r_j) dτ = - G ∫_b ρ₁ (3 r ^-5 r_i r_j - r^-3 δ_ij) dτ.

This unsuitable integral, obtained by deriving formally the potential gravific V₁₁ twice under the sign integral, converges since

|J_ij| ≤ 4 HAS G ∫_b ρ₁ r^α-3 D ³ R ≤ 16 π G has α^-3 ρ ₁ r_b^α

with the help of the condition of Hölder. Let us consider now in the vicinity of P occupying the position X a P_o point occupying the position X + H X _i^o, so as to this D (P, P_o) = |H| < r_b. Let us form the expression

K (H) = h^-1 + J_ij

for H ≠ 0. This integral is also convergent, as one can note it by using the same reasoning as for J_ij, and because X_j exists at the interior points. We have

K (H) = G ∫_b ρ₁ {h^-1 - (3 r ^-5 r_i r_j - r^-3 δ_ij)} D ³ R

with r_o ² = R ² + H ² - 2:00 r_i. We want to show that K (H) tends towards zero with H, independently of the position of P. But to arrive for this purpose, we must get rid of H in the denominator of the first term in the integral. By noticing that

r_o^-3 - r^-3 = ^-1 (R ² - r_o ²) (r_o^-2 + r^-2 + r_o^-1 r^-1) = H (2 r _i - H) ^-1 (r_o^-2 + r^-2 + r_o^-1 r^-1),

we find

h^-1 = - r_o^-3 δ_ij + h^-1 r_j (r_o^-3 - r^-3) = - r_o^-3 δ_ij + r_j (2 r _i - H) ^-1 (r_o^-2 + r^-2 + r_o^-1 r^-1).

For H = 0, this last expression are reduced to - r^-3 δ_ij + 3 r ^-5 r_i r_j, thus returning the intégrand of K (H) equal to zero and, consequently, K (0) = 0. If we can prove that the expression K (H) is continuous compared to H out of P, we will know that it is cancelled with H, and it will follow that the derivative of X_1j⁽¹⁾ compared to x_i exists out of P and is worth J_ij. Thus let us trace a small sphere of b_ε volume, radius r_ε < r_b and center P, so that |H| < r_ε. As P and P_o are all two outsides with the field B - b_ε, we can take |H| enough small so that the contribution K₂ (H) integration on B - b_ε is smaller in absolute value than ½ ε, where ε is a fixed positive number arbitrarily small. Now let us consider the contribution K₁ (H) of integration on b_ε, namely

K₁ (H) = G ∫_{b_ε} {- r_o^-3 δ_ij + r_j (2 r _i - H) ^-1 (r_o^-2 + r^-2 + r_o^-1 r^-1) - (3 r ^-5 r_i r_j - r^-3 δ_ij)} ρ₁ D ³ R .

By remembering that r_j and r_j - H δ_ij are the component j-ièmes (for I fixed) of the vectors PQ and P _o Q , respectively, we have

|r_j| ≤ R, |r_j - H δ_{ij| ≤ r_o, |2 r _i - H| ≤ |r_i| + |r_i - H| ≤ R + r_o.}

Thus, the expression between hooks is limited in absolute value by 4 r ^-3 + r^-2 r_o^-1 + r^-1 r_o^-2 + 2 r _o^-3. By using the condition of Hölder then, we obtain

K₁ (H) ≤ G has ∫_{b_ε} D ³ R ≤ 16πG has α^-1 r_ε^α + G has ∫_{b_ε} D ³ R

We have the right to suppose α < 1, because a condition of Hölder with a given exhibitor always implies a condition with a smaller positive exhibitor. However, in this case, the intégrand remaining in the integral of the member of right-hand side becomes infinite in two occasions, namely for R = 0 and r_o = 0. By considering the cases R separately < r_o and r_o < R, we have certainly

∫_{b_ε} D ³ R < ∫_{b_ε} 4 r ^α-3 D ³ R + ∫_{b'_ε} 4 r _o^α-3 D ³ R _o,

where b'_ε is a spheroidal field of ray 2r _ε centered on P_o. It follows that

|K₁ (H)| < 16πG (2+2^α) has α^-1 r_ε^α.

By choosing r_ε sufficiently small so that |K₁ (H)| < ½ ε, we have

|K (H)| < |K₁ (H)| + |K₁ (H)| < ε.

This proves that the function K (H) is continuous compared to H out of P, independently of the position of P. Donc, the existence of the derivative partial of the second order off V is established and, moreover:

∂_i ∂_j V (P) = - ∂_i X_j (P) = - ∂_j X_i (P).

In particular, the V₁₁ potential for which the density checks a Hölder condition and which is cancelled out of P is harmonic , since

∂_i ∂_i V₁₁ = J_ii = G ∫_b (3 r ^-5 R ² - 3 r ^-3) ρ₁ D ³ R ≡ 0.

Poisson's equation

If we superimpose on the distribution of density ρ₁ cancelling out of P another distribution of density ρ (P) constant everywhere in spherical volume B, we end to the result following valid for a distribution of continuous density in a spheroidal field satisfying to a condition of Hölder out of P: the derivative of gravity exist, and

∇ ² V (P) = - 4πG ρ (P),

where ∇ ² appoints the operator Laplacian (or simply the Laplacian ) ∂_i ∂_i. Finally, if we add the potentials with the distributions external with B, nothing is contributed to the Laplacian and the same equation above remains valid. This one is an partial derivative equation of the second order, which contains the equation of Laplace like a special case. It is known like Poisson's equation . It enables us to find the potential if we know the density and, reciprocally, to find a distribution of density if we know the potential. It is thus of an major importance in physical geodesy and, more precisely, for the problem which consists in finding the form of balance of a body in rotation. For this reason we took the trouble in this article to establish in detail and other the important property Poisson's equation of the Newtonian potential.

It is advisable to notice that we did not discuss here the situation prevailing in points of the border of the body. In these point-border, the potential of a distribution of density, like its derivative first, are continuous. On the other hand, on a border the derivative second of the potential do not exist in general. It is clear that they all cannot be continuous, because when we pass from a point external with an interior point through a border where the density is not cancelled, ∇ ² V undergoes a jump of - 4πGρ. A similar situation is carried out on surfaces inside a body on which the density is discontinuous. Then, the value of ∇ ² V jumps of - 4πG Δρ when one passes through the surface of discontinuity of density, where Δρ indicates the jump of density through this surface.

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