before us, the duangle and triangle, the relative motions correspond to certain parts only of that motion of the angle and triangle which would give us these centroids complete. The actual sequence of the motions of the pair is as follows:
So long as the point P, Fig. 98, moves towards U, T is the vertex of the angle on the arms of which PQ moves, the diameter
% = P_Q P Q
sin a
is the line T Q, and its half, V Q, the radius r,
sin 60 C
so that the arc Q U belongs to the larger, and Q W P to the smaller Cardanic circle. Further, as L UT Q = 60 and L P V Q = 120, the arc U Q is equal to the arc Q W P. From U on wards P moves upon the chord U Q to W, and Q along the half chord V T, this time Q has become the vertex of the angle along the arms of which
Fig. 97.
P Q slides; Q Vis the radius R and W Q the radius r, by which we obtain the arcs UT and Q V P. Proceeding from W y P moves next along the half chord W Q, while Q moves from T to P; U is now the vertex- angle, from which (with radius R} the arc Q T is described, on which again the curve Q W P rolls. After these motions P has reached the position Q, and vice versd, and the duangle has turned through an angle of 180. With its further rotation through two right angles the curve-triangle Q U T makes another complete revolution, and the duangle P V Q W one and a half revolutions. Thus, when the duangle has returned to its original position, the instantaneous centre has twice traversed the three sides of Q UT, and three times the two sides oiPVQ W,
