A Consequence of Mr. Cayley’s Theory of Skew Determinants, Concerning the Displacement of a Rigid System of an Even Number of Dimensions about a Fixed Origin

When we displace a rigid system of n orthogonal axes, represented in its primitive state by the identical substitution | 1 , 0 , 0 , ... 0 , 1 , 0 , ... ...................... | $$\left| \begin{gathered} 1,\quad 0,\quad 0,\quad ... \hfill \\ 0,\quad 1,\quad 0,\quad ... \hfill \\ ...................... \hfill \\ \end{gathered} \right|$$ about its origin to an infinitely small extent, the substitution representing its altered state in relation to the primitive axes cannot but assume the form | 1 , a , b , c , ... − a , 1 , f , g , ... − b , − f , 1 , h , ... ................................... | $$\left| \begin{gathered} 1,\quad a,\quad b,\quad c,\quad ... \hfill \\ - a,\quad 1,\quad f,\quad g,\quad ... \hfill \\ - b,\quad - f,\quad 1,\quad h,\quad ... \hfill \\ ................................... \hfill \\ \end{gathered} \right|$$ where a,b, ... denote infinitesimals of the first order, and squares and products of them are neglected ; for else the new system could not fulfil the orthogonal conditions. By erasing the units on the diagonal we should obtain the coefficients for the representation of the projections of the displacement of any point of the rigid system; should we then demand their vanishing in order to find resting points, a skew symmetrical determinant ought to vanish. But this is only then generally possible when n is odd; and as in this case the first minors do not vanish, there is consequently an axis, all points of which rest immovable during the displacement. Since such axis may be taken as a primitive one, and the corresponding dimension dropped, the question reduces to an even number of dimensions.