298 Proceedings of Royal Society of Edinburgh. [june 20 , 
one into some multiple of 7, the other into the corresponding multiple 
of 3 equal parts (actually into 21 and 9). Perpendiculars drawn 
from the points of section divide the diameters into graduated parts, 
representing the distances passed over by the end of the chain in 
equal portions of time during each of the separate simple oscilla- 
tions. Having completed the rhomboid ABCD, and divided it 
into a multitude of small rhomboids by parallels drawn through the 
divisions of its sides, we begin at the corner of any one of these, 
draw a line to the opposite corner, thence into the next, and so on, 
passing from side to side of the entire rhomboid, until we return to 
the first point. In this way we get an approximate representation 
of the path of the lower end of the chain when a plane oscillation 
(B) is imposed on a plane oscillation (A). 
But the chain may perform two simple oscillations (A) in different 
planes, the result being an elliptic movement ; and so also of the 
oscillation (B) ; and then the compound of the two (or rather four) 
must be got by carrying the centre of the one ellipse along the 
circumference of the other, in the manner used for the epicycloid. 
These curves present an endless diversity of form, according to 
the dimensions and relative positions of the ellipses. Adopting the 
ratio 7 : 3 for that of the periodic times, some of these are depicted 
in figure 4, a, b, e, d , e. In a and b the ellipses have been placed 
conformably and the curves are symmetric ; for a the motions were 
made both in one direction, and, as in the analogous case of the 
epicycloid, there are four , that is 7-3, lobes ; for b one of the 
motions has been reversed, and we find ten , that is 7 + 3, lobes ; 
c and d are corresponding examples with the axes of the ellipses set 
obliquely; while fur e the ellipse (B) is compressed into a straight 
line. These examples may give some faint idea of diversity of char- 
acter among the curves. 
While the lower end of the chain is describing some one of 
these curves, the points higher up are performing each its own peculiar 
evolution. As we ascend, the dimensions of the ellipse (B) decrease 
more rapidly than do those of (A), and consequently, along with 
its extent, the curve changes also its configuration ; and when we 
arrive at the height of the node B', the quicker ellipse has collapsed 
into a point, and the chain there describes simply the ellipse due 
to the oscillation (A). Above this height the ellipse (B) reappears, 
