of the Equinoxes. 61 
a spherical superficies to be described, passing through P, 
and let ADBC be the great circle of this sphere in the plane 
on which the whole is represented. Let the straight lines 
EF, HK pass through R, and be perpendicular to AB, DC 
respectively ; and let EPF, HPK be lesser circles of the 
sphere, EF being a diameter of the one, and HK a diameter 
of the other. Then it is evident, that by the simple motion 
of the body about AB only the particle P would move in the 
circumference FPE ; and by the simple motion of the body 
about DC only, P would move in the circumference HPK. 
Let the indefinitely small arcs Pj, Yq be those which P would 
describe in equal times with the revolutions about AB, DC 
separately, and let the parallelogram Yqps be completed on 
the spherical superficies. Then it is evident, from the com- 
position of motion, that the direction and velocity of P, in 
consequence of the compound motion, is as the diagonal P p of 
the parallelogram Yqps. 
Let RrMN be the orthographical projection of Yqps on the 
plane ADBC, and then as PR is perpendicular to this plane, 
it is evident that RrMN is a parallelogram, and that its dia- 
gonal RM is the direction and velocity of P in the projection, 
in consequence of the compound motion. It therefore follows, 
from article 3, as RN is the angular velocity about the axis 
AB, and Rr that about the axis DC, that RM is the angular 
velocity about the axis, round which the body is caused to 
revolve by the compound motion. 
6. The same things being supposed, and the parallelogram 
RrMN being the same in Fig. 4 as in Fig. 5, let RM pro- 
duced meet the circumference in L and Q, and the diameter 
TGS, at right angles to LQ, is the axis sought. 
The same axis may be obtained in the following manner. 
