Page:General Investigations of Curved Surfaces, by Carl Friedrich Gauss, translated into English by Adam Miller Hiltebeitel and James Caddall Morehead.djvu/51

 and from the formula

$$r\cos\psi = \tfrac{1}{2}\, \frac{\partial(r^{2})}{\partial q}$$

[3] $$\begin{alignedat}{2} r\cos\psi = q + \tfrac{2}{3}f^{\circ}p^{2}q & +\tfrac{1}{2}f'p^{3}q &&+ (\tfrac{2}{5}f'' - \tfrac{4}{45}{f^{\circ}}^{2})p^{4}q\quad\text{etc.} \\ & +\tfrac{3}{4}g^{\circ}p^{2}q^{2} &&+\tfrac{3}{5}g'p^{3}q^{2} \\ &&& +(\tfrac{4}{5}h^{\circ} - \tfrac{14}{45}{f^{\circ}}^{2})p^{2}q^{3} \end{alignedat}$$

These formulæ give the angle $\psi.$ In like manner, for the calculation of the angle $\phi,$  series for $r\cos\phi$  and $r\sin\phi$  are very elegantly developed by means of the partial differential equations

$$\begin{aligned} &\frac{\partial. r\cos\phi}{\partial p} = n\cos\phi. \sin\psi - r\sin\phi. \frac{\partial \phi}{\partial p} \\ &\frac{\partial. r\cos\phi}{\partial q} = \phantom{0}\cos\phi. \cos\psi - r\sin\phi. \frac{\partial \phi}{\partial q} \\ &\frac{\partial. r\sin\phi}{\partial p} = n\sin\phi. \sin\psi + r\cos\phi. \frac{\partial \phi}{\partial p} \\ &\frac{\partial. r\sin\phi}{\partial q} = \phantom{0}\sin\phi. \cos\psi + r\cos\phi. \frac{\partial \phi}{\partial q} \\ &n\cos\psi. \frac{\partial \phi}{\partial q} + \sin\psi. \frac{\partial \phi}{\partial p} = 0 \end{aligned}$$ A combination of these equations gives

$$\begin{alignedat}{3} &\frac{r\sin\psi}{n}. \frac{\partial. r\cos\phi}{\partial p} &&+ r\cos\psi. \frac{\partial. r\cos\phi}{\partial q} &&= r\cos\phi \\ &\frac{r\sin\psi}{n}. \frac{\partial. r\sin\phi}{\partial p} &&+ r\cos\psi. \frac{\partial. r\sin\phi}{\partial q} &&= r\sin\phi \end{alignedat}$$

From these two equations series for $r\cos\phi,$ $r\sin\phi$  are easily developed, whose first terms must evidently be $p,$  $q$  respectively. The series are

[4] $$\begin{alignedat}{3} r\cos\phi &= p + \tfrac{2}{3}f^{\circ}pq^{2} &&+ \tfrac{5}{12}f'p^{2}q^{2} &&+ (\tfrac{3}{10}f'' - \tfrac{8}{45}{f^{\circ}}^{2})p^{3}q^{2}\quad\text{etc.} \\ &&&+ \tfrac{1}{2}g^{\circ}pq^{3} &&+ \tfrac{7}{20}g'p^{2}q^{3} \\ &&&&&+ (\tfrac{2}{5}h^{\circ} - \tfrac{7}{45}{f^{\circ}}^{2})pq^{4} \end{alignedat}$$

[5] $ \begin{alignedat}{3} r\sin\phi &= q - \tfrac{1}{3}f^{\circ}p^{2}q &&- \tfrac{1}{6}f'p^{3}q &&- (\tfrac{1}{10}f'' - \tfrac{7}{90}{f^{\circ}}^{2})p^{4}q\quad\text{etc.} \\ &&&- \tfrac{1}{4}g^{\circ}p^{2}q^{2} &&- \tfrac{3}{20}g'p^{3}q^{2} \\ &&&&&- (\tfrac{1}{5}h^{\circ} + \tfrac{13}{90}{f^{\circ}}^{2})p^{2}q^{3} \end{alignedat}$

From a combination of equations [2], [3], [4], [5] a series for $r^{2}\cos(\psi + \phi),$ may be derived, and from this, dividing by the series [1], a series for $\cos(\psi + \phi),$  from