( 748 ) 
and so our proposition is proved. 
If in further consideration of the case of the surface of revolution 
we wish to see in what way 0 varies, we have to write down the 
differential equations giving the variability of the «’s and P’s. They 
now become : 
^■ = -.infa x a t sin(feoi V , 
da - 
—- = 4 nf a x a r, sin ip costp , 
dt 
"- = <* («!+«,) + /«* «•* 9> i 
We see that in — and — an equal constant term a («!+«,) = « R* k * 
dt 
appears. This means that the frequency n is modified by an amount 
of 2na jR 0 a h a . 
When we now differentiate according to t the relation 
we may arrive after some reduction at: 
from which it is evident, that the ellipse revolves with a constant 
angular velocity. 
These results agree quantitatively with those found by Prof. 
Korteweg. 
$ 26. The change in shape and position of the osculating curve 
does not seem to become simple for the general case n t = n x . 
Let us therefore restrict ourselves to k the case e 2 = 0; then the 
X&plane and the 7Z-plane are planes of symmetry for the surface. 
The first equation of (21) now becomes 
da x . 
— = - 4 nf a x a s sin ip cos ip. 
Or by introduction of l, : 
f = — 4 £ (l-o «. V ■ cos <p. 
