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

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 $n$  will be a function of $p,$  $q,$ such that for $q = 0$  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 $\theta$  denotes the angle between the element of this line and the element of the line for which $q$  is constant. Now since the axis of abscissas is itself a shortest line, and since, for it, we have everywhere $\theta = 0,$ we see that for $q = 0$  we must have everywhere

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

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

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

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

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