Page:DeSitterGravitation.djvu/4

Mar. 1911. Further, we put

The modulus of the transformation is $$q$$. We also introduce $$q_{1}=1-\sqrt{1-q^{2}}$$. If $$q$$ is a small quantity of the first order, then $$q_1$$ is of the second order. The cosines of the angles which the axis of the transformation makes with the axes of $$x, y, z$$ are denoted by $$\alpha, \beta, \gamma,$$ so that $$\alpha^2 + \beta^2 + \gamma^2 = 1$$.

The transformation-formulæ are then—

We find easily

and similarly for $$\eta'$$ and $$\zeta'$$.

Further,

In these formulæ $$r_q$$ and $$\phi_q$$ are the projections of $$\rho$$ and $$\phi$$ on the axis of transformation:—

If we put

we can easily verify that the transformation-formulas for $$(\xi), (\eta), (\zeta), (\kappa)$$ are the same as those for $$x, y, z, t$$, viz.—