(401) 
(we) = Acosacosn,r , § (y,+y,) = B cos b cos Ne pepe 2. 
The curve represented by (8) has therefore with respect to directions 
of chords parallel to the XY-plane as diameter a curve represented 
by the equations : 
G== Ay cos'n, ts 
oant, ir VEE (10) 
2, = C eos (n, + n,)t 
where 
A= ACOs Gl, B Beos b: 
To investigate the curves represented by (8) we can start from the 
simple curves represented by (10). In tig.3 such a curve is given 
perspectively, in fig. 4 (continuous lines) by projections for the case that 
n, and n, are commensurable and that we have n, = 2n, ; the twisted 
curve begins and ends in two vertices of the circumscribed parallelo- 
piped and is deseribed backwards and forwards. 
When a curve (10) is constructed we must bear in mind that 
Tit nO SUE TG Pt ES WEG yk Re, 
where 
Or Asin a, y= Bison b 
So we can think the curve (8) as described by a point moving 
along the curve (10) and vibrating at the same time according to 
the \- and Y-direction. 
From this we can see how the osculating curve changes for 
increasing values of « and 4, and we can make out when it shows 
double points. In fig. 4 the projections are represented (dotted lines) of 
an osculating curve for 7, == 2n, and small values of « and 4. 
§ 15. For the curves represented by (9) exists a simple method 
to construct the ZX- and ZY-projeetions, when the Y Y-projection 
is given. We can imagine / as even; the curves for odd values of / 
are the mirror image of the curves for even values of / with respect 
to the NZ-plane. The XY-projeetion is an entirely arbitrary Lissajous 
curve; for =O the projection of the point is on a diagonal of the 
circumscribed rectangle. 
Now however follows out of (9) 
z a“ Di 
‘ 
== GOS = ~ —— COs L 
7 aii B 
In fie. 5 is given how for every point on the XY-projection z is 
o T e e 
COS 
to be constructed. It is easy to show, that the points of intersection 
of (9) with the XN Y-plane lie on the ellipse : 
49 
Proceedings Royal Acad. Amsterdam. Vol. XIII. 
