Below follows the exact extract from Helmholtz's paper "On the conservation of force".
Let us now imagine, instead of the system $A$, a single material point $a$, it follows from what has been just proved, that the direction and magnitude of the force exerted by $a$ upon $m$ is only affected by the position which $m$ occupies with regard to $a$. But the only circumstance, as regards position, that can affect the action between the two points is the distance $ma$; the law, therefore, in this case would require to be so modified, that the direction and magnitude of the force must be functions of the said distance, which we shall name $r$. Let us suppose the co ordinates referred to any system of axes whatever whose origin lies in $a$, we have then $$md(q^2)=2Xdx+2Ydy+2Zdz=0 \tag{3}$$ as often as $$d(r^2)=2xdx+2ydy+2zdz=0$$ that is, as often as $$dz=-\frac{xdx+ydy}{z}$$ setting this value in equation (3), we obtain $$(X-\frac{x}{z}Z)dx+(Y-\frac{y}{z}Z)dy=0$$ for any values whatever of $dx$ and $dy$ ; hence also singly
$$X=\frac{x}{z}Z\tag{1}$$ $$Y=\frac{y}{z}Z\tag{2}$$ that is to say, the resultant must be directed towards the origin of coordinates, or towards the point $a$.
Hence in systems to which the principle of the conservation of force can be applied, in all its generality, the elementary forces of the material points must be central forces.
My question is about the equation (1) and (2). In which way these formulas reveal that the resultant must be directed towards the origin of coordinates?
The paper can be find here:
https://play.google.com/books/reader?id=C1i4AAAAIAAJ&hl=en&pg=GBS.PA120
