point m v fourth in the series of describing points, describes imme- diately right and left from 4 straight lines directed towards the centres C and D of the base circles 4 1 and 4 6> 3 , like the straight lines L m 2 and 3 m 3 from the homologous points m 2 and w? 3; joining each pair of straight lines is a circular arc described by the centre of the circle m 2 m 3 rolling in 1 2 , m B m 1 in 2 B , etc. This point-path is the common form for this series of curves.
The fifth point describes a four-cornered figure of elliptic arcs, in which loops make their appearance. This figure is shown on a doubled scale in Plate V., 2. In the point-path of Q the loops have separated and intersect each other, while with point 7, which is the centre M itself, the loops run over each other in a dumpy figure which in each period, or whole revolution of the element, traverses M three times. (See 23). The centre M of the square is also the centre of this point-path, which is the smallest of those which can be obtained by the motion of the curve triangle, or what we have called the concentral form of its point-path.
Plate VI. shows curves described by points on the prolongation through M l of the line on which were, the points just considered. Point 1 gives us a four-cornered figure consisting of elliptic arcs point 2 a straight-sided quadrilateral, covering part of the square ABCD, but having elliptically rounded corners; the points 3 and 4 elliptic quadrilaterals with concave sides. The last figure is shown to double the scale in Fig. 2; within it is the path for the point 5, which being a point upon the centroid gives us again a common form for this series of curves; it consists of four elliptic arcs with tangential prolongations at each cusp. The path 6 is described by the point which in Plate VI. , 1, coincides with the centre point M of the square; it is therefore the homocentral form of this series of curves; the point M, lastly, again gives us the concentral curve 7.
The point-paths 1', 2', and 3', shown in dotted lines, are examples of those described by points which do not lie in either of the three principal axes of the centroid.
