Page:General Investigations of Curved Surfaces, by Carl Friedrich Gauss, translated into English by Adam Miller Hiltebeitel and James Caddall Morehead.djvu/49

the solution of many important problems. But here we shall develop only a single example in order to show the nature of the method.

23.

We shall now consider the case where all the lines for which ${\textstyle p}$ is constant are shortest lines cutting orthogonally the line for which ${\textstyle \phi =0,}$ which line we can regard as the axis of abscissas. Let ${\textstyle A}$ be the point for which ${\textstyle r=0,}$ ${\textstyle D}$ any point whatever on the axis of abscissas, ${\textstyle AD=p,}$ ${\textstyle B}$ any point whatever on the shortest line normal to ${\textstyle AD}$ at ${\textstyle D,}$ and ${\textstyle BD=q,}$ so that ${\textstyle p}$ can be regarded as the abscissa, ${\textstyle q}$ the ordinate of the point ${\textstyle B.}$ The abscissas we assume positive on the branch of the axis of abscissas to which ${\textstyle \phi =0}$ corresponds, while we always regard ${\textstyle r}$ as positive. We take the ordinates positive in the region in which ${\textstyle \phi }$ is measured between ${\textstyle 0}$ and ${\textstyle 180^{\circ }.}$

By the theorem of Art. 16 we shall have

$$\omega =90^{\circ },\quad F=0,\quad G=1,$$

and we shall set also

$${\sqrt {E}}=n.$$

Thus ${\textstyle n}$ will be a function of ${\textstyle p,}$ ${\textstyle q,}$ such that for ${\textstyle q=0}$ it must become equal to unity. The application of the formula of Art. 18 to our case shows that on any shortest line whatever we must have

$$d\theta ={\frac {\partial n}{\partial q}}.dp,$$

where ${\textstyle \theta }$ denotes the angle between the element of this line and the element of the line for which ${\textstyle q}$ is constant. Now since the axis of abscissas is itself a shortest line, and since, for it, we have everywhere ${\textstyle \theta =0,}$ we see that for ${\textstyle q=0}$ we must have everywhere

$${\frac {\partial n}{\partial q}}=0.$$

Therefore we conclude that, if ${\textstyle n}$ is developed into a series in ascending powers of ${\textstyle q,}$ this series must have the following form:

$$n=1+fq^{2}+gq^{3}+hq^{4}+{\text{etc.}}$$

where ${\textstyle f,}$ ${\textstyle g,}$ ${\textstyle h,}$ etc., will be functions of ${\textstyle p,}$ and we set

$${\begin{alignedat}{4}f&=f^{\circ }&&+f'p&&+f''p^{2}&&+{\text{etc.}}\\g&=g^{\circ }&&+g'p&&+g''p^{2}&&+{\text{etc.}}\\h&=h^{\circ }&&+h'p&&+h''p^{2}&&+{\text{etc.}}\end{alignedat}}$$