where we may assume that
has the same sign as
then we have
It is evident that
denotes the angle between the cutting plane and another plane through this normal and that tangent which corresponds to the direction
Evidently, therefore,
takes its greatest (absolute) value, or
its smallest, when
and
its smallest absolute value, when
Therefore the greatest and the least curvatures occur in two planes perpendicular to each other. Hence these extreme values for
are
Their sum is
and their product
or the product of the two extreme radii of curvature is
This product, which is of great importance, merits a more rigorous development. In fact, from formulæ above we find
But from the third formula in [Theorem] 6, Art. 7, we easily infer that
therefore
Besides, from Art. 8,
therefore
Just as to each point on the curved surface corresponds a particular point
on the auxiliary sphere, by means of the normal erected at this point and the radius of