Center of Gravity of a System of Points with respect to a Point

Let $z_1, z_2, \ldots , z_n$ be arbitrary points lying in the finite part of the complex plane and let $m_1, m_2, \ldots , m_n$ be non-negative masses whose sum is $1$ which are placed at the points $z_1, z_2, \ldots , z_n$ respectively. In any linear transformation of the complex plane that carries the point at infinity into itself, the center of gravity $\zeta$ of such a mass-distribution is mapped by the same transformation, i.e. if we again place the point mass $m_{\nu }$ at the image-point $z_{\nu }'$ of $z_{\nu }$ ($1 \leq \nu \leq n$), then the center of gravity $\zeta '$ of this new mass distribution is identical with the image of the point $\zeta$.

In the sequel (102--156) we mean by a point'' of the complex plane an arbitrary finite point or the point at infinity, by a circle'' the circumference of a circle or a straight line---the latter passes through the point at infinity---and by a ``circular domain'' a domain whose boundary is a circle. A circular domain is either the closed interior or the closed exterior of a circle or a closed half-plane, according as it does not contain the point at infinity or contains it in its interior or on its boundary.

Let the points $z_1, z_2, \ldots , z_n, z$ and the masses $m_1, m_2, \ldots , m_n$ be given. Let $z$ be different from all the points $z_1, z_2, \ldots , z_n$ while of the latter some may coincide. Let $m_1 \geq 0, m_2 \geq 0, \ldots , m_n \geq 0, m_1+m_2+\cdots +m_n=1$.

Find a point $\zeta$ situated in such a manner that, in a linear transformation of the plane that maps the $n+2$ points

into the points

respectively, $\zeta '$ becomes the center of gravity of the mass-distribution determined by the masses $m_{\nu }$ placed at the points $z_{\nu }'$ ($\nu =1,2,3,\ldots ,n$).

There are infinitely many linear transformations that map $z$ into $\infty$; the point $\zeta$ however is independent of the choice of the underlying transformation.

The point $\zeta =\zeta_z$, defined above, which is determined uniquely by the system of points $z_1, z_2, \ldots , z_n$, by the masses $m_1, m_2, \ldots , m_n$, $m_1+m_2+\cdots +m_n=1$, placed there, and by the ``point of reference'' $z$ which must be distinct from all the points $z_{\nu }$, is called the center of gravity of the mass distribution $m_1, m_2, \ldots , m_n$ with respect to $z$. If $z$ is the point at infinity, $z=\infty$, then $\zeta_z=\zeta_{\infty }$ is the ordinary center of gravity.

Consider all possible mass distributions with total mass $1$ over the fixed points $z_1, z_2, \ldots , z_n$ and let $z$ be a point distinct from $z_1, z_2, \ldots , z_n$. The centers of gravity $\zeta_z$ of all mass distributions of this kind with respect to $z$ cover a ``circular-arc polygon'' $\mathfrak {C}_z$.

Let $\mathfrak {C}_z$ be the circular-arc polygon defined in 103 and $w_1, w_2$ two points of $\mathfrak {C}_z$. Consider the circle through $w_1, w_2$ and $z$. The arc of this circle with end-points $w_1, w_2$, that does not contain $z$, is contained in $\mathfrak {C}_z$, in other words $\mathfrak {C}_z$ is circularly-convex with respect to $z$. (We call $\mathfrak {C}_z$ the smallest circularly-convex domain with respect to $z$ that contains the points $z_1, z_2, \ldots , z_n$.)

If the points $z_1, z_2, \ldots , z_n$ all belong to a circular domain $C$ and if the point $z$ lies outside $C$, then the circular-arc polygon $\mathfrak {C}_z$ [103] lies completely inside $C$.

Let $z_1, z_2, \ldots , z_n$ be arbitrary points and $z$ a point distinct from these. Let the same mass $1/n$ be placed at each of the points $z_{\nu }$. In the sequel we consider only the center of gravity $\zeta =\zeta_z$ of this special mass distribution and call it for brevity the center of gravity of the points $z_1, z_2, \ldots , z_n$ with respect to $z$.

Let $\zeta_z$ be the center of gravity of $z_1, z_2, \ldots , z_n$ with respect to $z$. Every circle through $z$ and $\zeta_z$ separates the points $z_1, z_2, \ldots , z_n$, i.e. either both circular domains bounded by the circle contain some of the points $z_1, z_2, \ldots , z_n$ in their interior or all the points $z_1, z_2, \ldots , z_n$ lie on the circle.

If all the points $z_1, z_2, \ldots , z_n$ belong to a circular domain $C$, then the points $z$ and $\zeta_z$ cannot both lie outside $C$. If one of them, e.g. $z$, lies outside $C$, then the other, $\zeta_z$, lies in the interior of $C$, except in the case where all the points $z_1, z_2, \ldots , z_n$ coincide at a boundary point of $C$ and $\zeta_z$ then also lies there.

Let the fixed points $w_1, w_2, \ldots , w_k, z$ be pair-wise distinct. Let the complex numbers $z_1, z_2, \ldots , z_n$ take only the $k$ different values $w_1, w_2, \ldots , w_k$. The centers of gravity with respect to $z$ of all systems $z_1, z_2, \ldots , z_n$ of this kind lie everywhere dense in the smallest domain, circularly convex with respect to $z$, that contains the points $w_1, w_2, \ldots , w_k$.

The points $z_1, z_2, \ldots , z_n$ are fixed and the variable point $z$ converges to $z_1$. To what limiting position does the center of gravity $\zeta_z$ of the points $z_1, z_2, \ldots , z_n$ with respect to $z$ tend?

The center of gravity $\zeta_z$ of the finite fixed points $z_1, z_2, \ldots , z_n$ with respect to the variable point $z$ may for sufficiently large $z$ be expanded in decreasing powers of $z$. Calculate the first two terms of the expansion.