The mirror can compensate the tested mirror very well in the optical manufacturing, which with some difficult problems can be solved in the manufacturing. Based on the third-order aberration theory the initial configuration parameters of the mirror compensator in testing the conicoid are solved in theory, making spherical aberration coefficient ∑S1=0, and the relations between the compensating mirror and the tested mirror are given, with the surface of compensator chosen as sphere or ellipsoid. Setting e21=0,0.1,0.2,0.3,0.4, the graph of e22/α～β, α r01/r02～β is plotted based on the equations, the relations of β and α, e21, e22, r01, r02, d12 and the initial configuration parameters of the compensating mirror are also gotten. There are two cases about the compensator: one is α>0, α>1, and the compensator is set in front of the curvature center of the tested mirror; the other is α<0, and the compensator is set in back of the curvature center of the tested mirror,both including β>0, β=0, β<0. By analysing in detail, it is found that both the convex and the concave conicoid mirrors can be compensated and tested by mirror compensators, all cases of compensations and tests are included, some have been brought forward, but others have not, which benifits the general analysis of the compensation and the test.