259 
change of a during the course of each revolution, The curve 
described, instead of re-entering exactly at the time mn, does so 
nearly but not accurately, owing to the small change in a, so 
that it starts on a slightly different course after each interval 
mn. So that the appearance is that of a curve, slowly changing 
its character and position, until, if 7, and r,are commensurable, 
it finally returns to its primitive state. If 7, and 7, are not 
commensurable it never does so. 
All these points can be clearly indicated by fixing a screen 
so as to cover the disks: having in it two slits at right angles to 
each other, one in which P, and P, slide, and the other in which 
Q,, Q, slide. Itis manifest that any point in this screen will 
trace out the curve 
2=asin("t+a), y=bsin(t+8), 
T; T 2 
when a and } can be arranged at pleasure by shifting the pegs, 
tT, 7, by shifting the band, and a, B by arranging the position 
of the disks at starting. A pencil pressed against the screen 
will trace out the curves. 
The form of the curves may also be represented to the eye 
by piercing a hole in the screen, and placing a strong light 
behind it; or by means of the electric spark passing between 
the ends of fine wires, the ends being fixed close together at any 
point on the screen. 
Another way. of representing the curve to the eye is this: 
two screens are provided, one is attached to the disks B,, B, by 
the pegs P,, P,; P,, P, do not slide in a slit, but there is a slit 
extending nearly from P, to P,. Similarly in the screen 
attached to C,, C, there is a slit extending nearly between 
Q, and Q, Light if placed behind can be seen through both 
screens at once only at the point where the slits cross, namely, 
at the point P, whose locus is the point we have been treating of. 
20—2 
