(4)  Redshift-magnitude  diagram

Redshift- magnitude  relation is

m = M + 5lg r + 40

where M is absolute magnitude. In the computation we use median absolute magnitude  M_v ,  I adopt

                  for the galaxy

                  for the  QSO^[5] 

therefore

          and         

When  Z<2.3,  it must make use of formulae inside the sphere to compute, that is formulae (21), (22) and (13);  Z>2.3 is by mean of formulae outside the sphere, that is, formulae (21), (23) and (13).