On the resultant of forces (Q1897519)

Let \(T\) be a binary operation on \(\mathbb{R}^n\) having the properties: (1) \(T\) is commutative and associative; (2) \(aTb=a+b\) when \(b\) is a scalar multiple of \(a\); (3) \((Aa) T(Ab)=A(aTb)\) for an orthogonal transformation with determinant 1. In 1769 d'Alembert proved that for \(n=3\) the conditions (1)--(3) imply that \(aTb=a+b\) for all \(a,b\) if we assume that \(T\) is continuous. d'Alembert used this theorem to prove that the resultant of forces is obtained by the vectorial sum of the components. In this paper the author shows that when \(n \geq 3\), conditions (1)--(3) imply that (3) holds true for any orthogonal transformation; and more generally, he describes all operations \(T\) on \(\mathbb{R}^n\) having these three properties. He proves that for \(n \geq 3\) the assumption on the continuity in d'Alembert's result can be replaced by a weaker condition, namely: there is a \(\delta > 0\) such that the set \(\{aTb : |a |=|b |< \delta\}\) is not everywhere dense in \(\mathbb{R}^n\). In the case \(n=2\) the situation is quite different, and a serious study of this case is given, too.