Page:General Investigations of Curved Surfaces, by Carl Friedrich Gauss, translated into English by Adam Miller Hiltebeitel and James Caddall Morehead.djvu/39

and consequently

$$AC+CB'=AC+BC.\cos \omega =AB-BC.(1-\cos \omega )=AB'-BC.(1-\cos \omega ),$$

i.e., the path from ${\textstyle A}$ to ${\textstyle B'}$ through the point ${\textstyle C}$ is shorter than the shortest line, Q.E.A.

16.

With the theorem of the preceding article we associate another, which we state as follows: If on a curved surface we imagine any line whatever, from the different points of which are drawn at right angles and toward the same side an infinite number of shortest lines of the same length, the curve which joins their other extremities will cut each of the lines at right angles. For the demonstration of this theorem no change need be made in the preceding analysis, except that ${\textstyle \phi }$ must denote the length of the given curve measured from an arbitrary point; or rather, a function of this length. Thus all of the reasoning will hold here also, with this modification, that ${\textstyle S=0}$ for ${\textstyle r=0}$ is now implied in the hypothesis itself. Moreover, this theorem is more general than the preceding one, for we can regard it as including the first one if we take for the given line the infinitely small circle described about the centre ${\textstyle A.}$ Finally, we may say that here also geometric considerations may take the place of the analysis, which, however, we shall not take the time to consider here, since they are sufficiently obvious.

17.

We return to the formula

$${\sqrt {E\,dp^{2}+2F\,dp.dq+G\,dq^{2}}},$$

which expresses generally the magnitude of a linear element on the curved surface, and investigate, first of all, the geometric meaning of the coefficients ${\textstyle E,}$ ${\textstyle F,}$ ${\textstyle G.}$ We have already said in Art. 5 that two systems of lines may be supposed to lie on the curved surface, ${\textstyle p}$ being variable, ${\textstyle q}$ constant along each of the lines of the one system; and ${\textstyle q}$ variable, ${\textstyle p}$ constant along each of the lines of the other system. Any point whatever on the surface can be regarded as the intersection of a line of the first system with a line of the second; and then the element of the first line adjacent to this point and corresponding to a variation ${\textstyle dp}$ will be equal to ${\textstyle {\sqrt {E}}.dp,}$ and the element of the second line corresponding to the variation ${\textstyle dq}$ will be equal to ${\textstyle {\sqrt {G}}.dq.}$ Finally, denoting by ${\textstyle \omega }$ the angle between these elements, it is easily seen that we shall have

$$\cos \omega ={\frac {F}{\sqrt {EG}}}.$$