( w ) 
The ellipses are described in rectangles having their sides parallel 
to the axes and whose vertices, as is evident from 
constant, 
lie on the circumference of a circle. 
To investigate the change in shape and position we may write 
down the well-known relations which may serve for the calculation 
ot the axes of the ellipse and the angle of inclination of the long 
axis with the Jf-axis. If Ah and Bh are half the laiger and half 
the smaller axis and if 6 is the angle in view, then these relations 
U) 
A’B'h* ~ ff 1 a, sin* <p ..( 2 ) 
hrom (1) and (2) we now deduce at once: The sum of the 
squares of the axes of the ellipse is constant. 
$ 25. From what we have just found we can easily prove that 
in case the surface is a surface of revolution the osculating ellipse 
has an invariable shape. 
Then namely we find: 
where: 
— R _ £ a K-f a,)* 4- f «, sin* <p, 
«=**+*, 
4n< ^ 8g* ’ 
/= — ~ T -7~* * 
2n* 4g* 
2 a («i + «*)* nn* <p — constant, 
«, + a, = constant, 
a t sin * <p = constant. 
From (2) it then follows that 
ABk * == constan 
from which in connection with the close of $ 24 we may 
elude that 
