Page:General Investigations of Curved Surfaces, by Carl Friedrich Gauss, translated into English by Adam Miller Hiltebeitel and James Caddall Morehead.djvu/61

three rectangular coordinates, the simplest method is to express one coordinate as a function of the other two. In this way we obtain the simplest expression for the measure of curvature. But, at the same time, there arises a remarkable relation between this measure of curvature and the curvatures of the curves formed by the intersections of the curved surface with planes normal to it. Euler, as is well known, first showed that two of these cutting planes which intersect each other at right angles have this property, that in one is found the greatest and in the other the smallest radius of curvature; or, more correctly, that in them the two extreme curvatures are found. It will follow then from the above mentioned expression for the measure of curvature that this will be equal to a fraction whose numerator is unity and whose denominator is the product of the extreme radii of curvature. The expression for the measure of curvature will be less simple, if the nature of the curved surface is determined by an equation in ${\textstyle x,}$ ${\textstyle y,}$ ${\textstyle z.}$ And it will become still more complex, if the nature of the curved surface is given so that ${\textstyle x,}$ ${\textstyle y,}$ ${\textstyle z}$ are expressed in the form of functions of two new variables ${\textstyle p,}$ ${\textstyle q.}$ In this last case the expression involves fifteen elements, namely, the partial differential coefficients of the first and second orders of ${\textstyle x,}$ ${\textstyle y,}$ ${\textstyle z}$ with respect to ${\textstyle p}$ and ${\textstyle q.}$ But it is less important in itself than for the reason that it facilitates the transition to another expression, which must be classed with the most remarkable theorems of this study. If the nature of the curved surface be expressed by this method, the general expression for any linear element upon it, or for ${\textstyle {\sqrt {dx^{2}+dy^{2}+dz^{2}}},}$ has the form ${\textstyle {\sqrt {E\,dp^{2}+2F\,dp.dq+G\,dq^{2}}},}$ where ${\textstyle E,}$ ${\textstyle F,}$ ${\textstyle G}$ are again functions of ${\textstyle p}$ and ${\textstyle q.}$ The new expression for the measure of curvature mentioned above contains merely these magnitudes and their partial differential coefficients of the first and second order. Therefore we notice that, in order to determine the measure of curvature, it is necessary to know only the general expression for a linear element; the expressions for the coordinates ${\textstyle x,}$ ${\textstyle y,}$ ${\textstyle z}$ are not required. A direct result from this is the remarkable theorem: If a curved surface, or a part of it, can be developed upon another surface, the measure of curvature at every point remains unchanged after the development. In particular, it follows from this further: Upon a curved surface that can be developed upon a plane, the measure of curvature is everywhere equal to zero. From this we derive at once the characteristic equation of surfaces developable upon a plane, namely,

$${\frac {\partial ^{2}z}{\partial x^{2}}}\cdot {\frac {\partial ^{2}z}{\partial y^{2}}}-\left({\frac {\partial ^{2}z}{\partial x.\partial y}}\right)^{2}=0,$$

when ${\textstyle z}$ is regarded as a function of ${\textstyle x}$ and ${\textstyle y.}$ This equation has been known for some time, but according to the author’s judgment it has not been established previously with the necessary rigor.