( 744 ) 
For the surface the lowest point is an umbilical point. To this 
belongs as special case the surface of revolution with the Z-axis as 
axis of revolution, which case is treated by Prof. Korteweg at the 
close of his treatise quoted before. 
Omitting the terms of higher order than h\ because in the equations 
of motion we admit no terms of higher order than h*, and omitting 
the terms of order A% because in the equations of motion no terms 
of order h * can be disturbing, we may write the equation of the 
surface: 
+ w* + + e * a *y + 4 - e 2 + ^ 4 )> 
where we avail ourselves of the fact, that by means of a rotation 
of the system of coordinates round the Z-axis the coefficients of 
xy * and x*y may be rendered equal. 
The solution at first approximation is: 
« = ^ cos (i nt -f- 2n&), 
y-^—c°s{nt -f 2n&); 
where n = l/2c 1 —1/2c # . 
§ 21. Let us now pass to the simplification of the equations of 
motion. Corresponding to what was said in §15 for the case S =4 
we may here replace in the terms of order h 3 of the equations of 
motion: 
x* by a x — n'x z , 
y* » — n V» 
y » - 
The equations become: 
x-\. n *x+Ae 1 a; i -\-Ze i x*y+2e l a:y*+e l y* + ^ (s’-hf) *=°* I 
Here we may omit no terms, for all the terms of order A* are 
disturbing. The equations may be written as follows: 
