Page:General Investigations of Curved Surfaces, by Carl Friedrich Gauss, translated into English by Adam Miller Hiltebeitel and James Caddall Morehead.djvu/50

 or

$$\begin{alignedat}{2} n = 1 + f^{\circ}q^{2} &+ f'pq^{2}  &&+ f''p^{2}q^{2} + \text{etc.} \\ &+ g^{\circ} q^{3} &&+ g'pq^{3} + \text{etc.} \\ &            &&+ h^{\circ}q^{4} + \text{etc. etc.} \end{alignedat}$$

24.

The equations of Art. 22 give, in our case,

$$\begin{aligned} &n\sin\psi = \frac{\partial r}{\partial p},\quad \cos\psi = \frac{\partial r}{\partial q},\quad -n\cos\psi = m. \frac{\partial \phi}{\partial r},\quad \sin\psi = m. \frac{\partial \phi}{\partial q}, \\ &n^{2} = n^{2}\left(\frac{\partial r}{\partial q}\right)^{2} + \left(\frac{\partial r}{\partial p}\right)^{2},\quad n^{2}. \frac{\partial r}{\partial q}. \frac{\partial \phi}{\partial q}     + \frac{\partial r}{\partial p}. \frac{\partial \phi}{\partial p} = 0 \end{aligned}$$

By the aid of these equations, the fifth and sixth of which are contained in the others, series can be developed for $r,$  $\phi,$  $\psi,$  $m,$ or for any functions whatever of these quantities. We are going to establish here those series that are especially worthy of attention.

Since for infinitely small values of $p,$  $q$ we must have

$$r^{2} = p^{2} + q^{2},$$

the series for $r^{2}$ will begin with the terms $p^{2} + q^{2}.$  We obtain the terms of higher order by the method of undetermined coefficients, by means of the equation

$$\left(\frac{1}{n} . \frac{\partial(r^{2})}{\partial p}\right)^{2} + \left(\frac{\partial(r^{2})}{\partial q}\right)^{2} = 4r^{2}$$

Thus we have [1] $\displaystyle\begin{alignedat}{3} r^{2} &= p^{2} + \tfrac{2}{3}f^{\circ}p^{2}q^{2} &&+ \tfrac{1}{2}f'p^{3}q^{2} &&+ (\tfrac{2}{5}f'' - \tfrac{4}{45}{f^{\circ}}^{2})p^{4}q^{2}\quad\text{etc.} \\ &+ q^{2} &&+ \tfrac{1}{2}g^{\circ}p^{2}q^{3} &&+ \tfrac{2}{5}g'p^{3}q^{3} \\ &&& &&+(\tfrac{2}{5}h^{\circ} - \tfrac{7}{45}{f^{\circ}}^{2})p^{2}q^{4} \end{alignedat}$

Then we have, from the formula

$$r\sin\psi = \frac{1}{2n}. \frac{\partial(r^{2})}{\partial p},$$

[2] $\displaystyle\begin{alignedat}{2} r\sin\psi = p - \tfrac{1}{3}f^{\circ}pq^{2} & -\tfrac{1}{4}f'p^{2}q^{2} &&- (\tfrac{1}{5}f'' + \tfrac{8}{45}{f^{\circ}}^{2})p^{3}q^{2}\quad\text{etc.} \\ & -\tfrac{1}{2}g^{\circ}pq^{3} &&-\tfrac{2}{5}g'p^{2}q^{3} \\ &&& -(\tfrac{3}{5}h^{\circ} - \tfrac{8}{45}{f^{\circ}}^{2})pq^{4} \end{alignedat}$