If we look out from any point in an isotropic, homogeneous Universe it must appear to expand in the same way. This expansion, mentioned in Section 21, has the form H(t)x with H = Ṙ/R when the coordinate system is chosen to coincide with a point of zero systematic motion.

To see this pretend to be a ‘fundamental observer’ at point O, moving with the average flow. Looking out at time t to an arbitrary point P along the position vector x = OP, you see the velocity v(x, t) of P relative to you. Another fundamental observer is moving with the flow at O′ and his velocity relative to you is v(s, t) where s = OO′. (The reader may find it helpful to draw a diagram.) He measures the velocity of the same point P, located at x′ = x - s in his coordinate system, and finds it to be v′(x′, t) = v′(x - s, t) = v(x, t) - v(s, t). Now, since the Universe is homogeneous and isotropic, v′ must be the same function of x′ and t that v is of x and t. Therefore, v(x - s, t) = v(x, t) - v(s, t). By inspection the solution of this functional equation is that the velocity is a linear function of position, so it has the form v = f(t)x = Ẉ.