Page:General Investigations of Curved Surfaces, by Carl Friedrich Gauss, translated into English by Adam Miller Hiltebeitel and James Caddall Morehead.djvu/23

In the first method, such a criterion is to be drawn from the sign of the quantity ${\textstyle W.}$ Indeed, generally speaking, the curved surface divides those regions of space in which ${\textstyle W}$ keeps a positive value from those in which the value of ${\textstyle W}$ becomes negative. In fact, it is easily seen from this theorem that, if ${\textstyle W}$ takes a positive value toward the exterior region, and if the normal is supposed to be drawn outwardly, the first solution is to be taken. Moreover, it will be easy to decide in any case whether the same rule for the sign of ${\textstyle W}$ is to hold throughout the entire surface, or whether for different parts there will be different rules. As long as the coefficients ${\textstyle P,}$ ${\textstyle Q,}$ ${\textstyle R}$ have finite values and do not all vanish at the same time, the law of continuity will prevent any change.

If we follow the second method, we can imagine two systems of curved lines on the curved surface, one system for which ${\textstyle p}$ is variable, ${\textstyle q}$ constant; the other for which ${\textstyle q}$ is variable, ${\textstyle p}$ constant. The respective positions of these lines with reference to the exterior region will decide which of the two solutions must be taken. In fact, whenever the three lines, namely, the branch of the line of the former system going out from the point ${\textstyle A}$ as ${\textstyle p}$ increases, the branch of the line of the latter system going out from the point ${\textstyle A}$ as ${\textstyle q}$ increases, and the normal drawn toward the exterior region, are similarly placed as the ${\textstyle x,}$ ${\textstyle y,}$ ${\textstyle z}$ axes respectively from the origin of abscissas (e.g., if, both for the former three lines and for the latter three, we can conceive the first directed to the left, the second to the right, and the third upward), the first solution is to be taken. But whenever the relative position of the three lines is opposite to the relative position of the ${\textstyle x,}$ ${\textstyle y,}$ ${\textstyle z}$ axes, the second solution will hold.

In the third method, it is to be seen whether, when ${\textstyle z}$ receives a positive increment, ${\textstyle x}$ and ${\textstyle y}$ remaining constant, the point crosses toward the exterior or the interior region. In the former case, for the normal drawn outward, the first solution holds; in the latter case, the second.

6.

Just as each definite point on the curved surface is made to correspond to a definite point on the sphere, by the direction of the normal to the curved surface which is transferred to the surface of the sphere, so also any line whatever, or any figure whatever, on the latter will be represented by a corresponding line or figure on the former. In the comparison of two figures corresponding to one another in this way, one of which will be as the map of the other, two important points are to be considered, one when quantity alone is considered, the other when, disregarding quantitative relations, position alone is considered.

The first of these important points will be the basis of some ideas which it seems judicious to introduce into the theory of curved surfaces. Thus, to each part of a curved