( 621 ) 
These equations are in general sufficient to arrive at the solution 
at first approximation. This then becomes: 
x = Ahcos{ ni t + X), | 
( 4 ) 
where n 1 =V' 2 c lt n, = V 2c,. 
Here Ah, Bh, X and p are constants of integration; we suppose 
A and B to be of moderate greatness. 
At first approximation therefore the horizontal projection of the 
moving point describes a Lissajous curve, which is closed when 
mi x = qn„ where p and q are integers. If pn x = qn t + t^ e curve 
described is not closed, but it consists of a succession of parts each 
of which differs but little from a closed curve. These last closed 
curves have however various shapes which answer to different values 
of the difference in phase. They are all described in the rectangle 
with 2 Ah and 2Bh as sides. 
§ 3. If we wish to take into consideration the terms of a higher 
order appearing in (2) we generally have but to apply small modi¬ 
fications to the first approximation. 
These modifications are, however, not small in case a relation 
exists of the form: 
pn 1 — qn t -\-Q; 
where SsTp -f g< 4 and — is very small (what is meant here by 
“very small” will be evident later on). 
When by applying the method of consecutive approximations, 
starting from (A) as first approximation, we try to find expansions 
in series for x and y, we shall find, when substituting the expres¬ 
sions (4) into the terms of higher order of (2) and developing 
the products and powers of the cosines, in case ~ is very small, 
periodical terms which have about the same period as the principal 
vibration, to which the equation in which the indicated term appears 
relates more especially. Such terms in the equations of motion give 
rise in the expansions in series for x and y to terms with abnor¬ 
mally great amplitude. These amplitudes may reach the order A and 
even seem to be greater still. 
This proves that in the case supposed our first approximation was 
not correct. It is evident that in the equations of motion there are 
terms of higher order, which are of influence even on the first 
