( 627 ) 
where « 2 , (?, and f? a are slowly variable; for — and 2 are 
dt dt 
of order h\ ~ and ^ of order h. (Comp. (9)). 
For every arbitrary moment the «’s and then’s have a definite value. 
These values determine a certain Lissajous curve. This curve we 
shall call the osculating curve for the moment indicated, which 
name is in use in the theory of disturbances. (See among others 
H. Poincaris, Lec^ons de mecanique celeste, vol. I, page 90). Thus 
in our problem the osculating curves are the wellknown Lissajous 
figures for 2 octaves. 
By the change of the origin of time we may write the equations 
of an osculating curve : 
y = ^i2 0 /i l/l—geos (2« 1 <—y) ; 
where as in § 5 we have introduced £ instead of a l and « 2 ; here 
too <p means 4ra 1 (&— &). 
We now see that <p is the value of the difference in phase, to 
which the osculating curve corresponds when the phase is calculated 
from the moment of the greatest deviation to the right. 
The amplitudes of the x- and y-vibrations being respectively 
R a h)/£ and \R 0 h j/1 — £, the vertices of the rectangles, in which the 
osculating curves are described lie on the circumference of an ellipse 
with its great axis along the a?-axis and having a length of 
2 RJi, and its small axis along the y-axis and having a length 
of RJi. 
Now £ changes its value between £ x and £ s , so the rectangles in 
which the osculating curves are described also lie between two 
extremes. 
Moreover as according to (12) to each value of £ a value of cos <p 
belongs all osculating curves may now be constructed. 
It follows from (13) that for the extreme values of £ we find 
sinrf = 0; so in the extreme rectangles parabolae are described. 
The distance from OX of the node of an arbitrary osculating 
curve is — RfE from which it is evident that the nodes and 
2£ 
also the vertices of the parabolae lie all on the same side of 0 
lying below 0 for positive values of K (see fig. 2). 
Envelope of the osculating curves. 
$ 8. If we perform the elimination of t and <p from: 
42 
Proceedings Royal Acad. Amsterdam. Vol. XII. 
