Page:General Investigations of Curved Surfaces, by Carl Friedrich Gauss, translated into English by Adam Miller Hiltebeitel and James Caddall Morehead.djvu/22

Substituting these values in the formula given above, we obtain

$$(aX+bY+cZ)\,dp+(a'X+b'Y+c'Z)\,dq=0$$

Since this equation must hold independently of the values of the differentials ${\textstyle dp,}$ ${\textstyle dq,}$ we evidently shall have

$$aX+bY+cZ=0,\quad a'X+b'Y+c'Z=0$$

From this we see that ${\textstyle X,}$ ${\textstyle Y,}$ ${\textstyle Z}$ will be proportioned to the quantities

$$bc'-cb',\quad ca'-ac',\quad ab'-ba'$$

Hence, on setting, for brevity,

$${\sqrt {(bc'-cb')^{2}+(ca'-ac')^{2}+(ab'-ba')^{2}}}=\Delta$$

we shall have either

$${\begin{aligned}X&={\frac {bc'-cb'}{\Delta }},\quad &Y&={\frac {ca'-ac'}{\Delta }},\quad &Z&={\frac {ab'-ba'}{\Delta }}\end{aligned}}$$

or

$${\begin{aligned}X&={\frac {cb'-bc'}{\Delta }},\quad &Y&={\frac {ac'-ca'}{\Delta }},\quad &Z&={\frac {ba'-ab'}{\Delta }}\end{aligned}}$$

With these two general methods is associated a third, in which one of the coordinates, ${\textstyle z,}$ say, is expressed in the form of a function of the other two, ${\textstyle x,}$ ${\textstyle y.}$ This method is evidently only a particular case either of the first method, or of the second. If we set

$$dz=t\,dx+u\,dy$$

we shall have either

$${\begin{aligned}X&={\frac {-t}{\sqrt {1+t^{2}+u^{2}}}},&Y&={\frac {-u}{\sqrt {1+t^{2}+u^{2}}}},&Z&={\frac {1}{\sqrt {1+t^{2}+u^{2}}}}\end{aligned}}$$

or

$${\begin{aligned}X&={\frac {t}{\sqrt {1+t^{2}+u^{2}}}},&Y&={\frac {u}{\sqrt {1+t^{2}+u^{2}}}},&Z&={\frac {-1}{\sqrt {1+t^{2}+u^{2}}}}\end{aligned}}$$

5.

The two solutions found in the preceding article evidently refer to opposite points of the sphere, or to opposite directions, as one would expect, since the normal may be drawn toward either of the two sides of the curved surface. If we wish to distinguish between the two regions bordering upon the surface, and call one the exterior region and the other the interior region, we can then assign to each of the two normals its appropriate solution by aid of the theorem derived in Art. 2 (VII), and at the same time establish a criterion for distinguishing the one region from the other.