Page:General Investigations of Curved Surfaces, by Carl Friedrich Gauss, translated into English by Adam Miller Hiltebeitel and James Caddall Morehead.djvu/52

 which may be found a series for the angle $\psi + \phi$ itself. However, the same series can be obtained more elegantly in the following manner. By differentiating the first and second of the equations introduced at the beginning of this article, we obtain

$$\sin\psi. \frac{\partial n}{\partial q} + n\cos\psi. \frac{\partial \psi}{\partial q} +  \sin\psi. \frac{\partial \psi}{\partial p} = 0$$

and this combined with the equation

$$n\cos\psi. \frac{\partial \phi}{\partial q} + \sin\psi. \frac{\partial \phi}{\partial p} = 0$$

gives

$$\frac{r\sin\psi}{n}. \frac{\partial n}{\partial q} + \frac{r\sin\psi}{n}. \frac{\partial(\psi + \phi)}{\partial p} + r\cos\psi. \frac{\partial(\psi + \phi)}{\partial q} = 0$$

From this equation, by aid of the method of undetermined coefficients, we can easily derive the series for $\psi + \phi,$ if we observe that its first term must be $\frac{1}{2}\pi,$  the radius being taken equal to unity and $2\pi$  denoting the circumference of the circle,

[6] $\displaystyle\begin{alignedat}{2} \psi + \phi = \tfrac{1}{2}\pi - f^{\circ}pq &- \tfrac{2}{3}f'p^{2}q &&- (\tfrac{1}{2}f'' - \tfrac{1}{6}{f^{\circ}}^{2})p^{3}q\quad\text{etc.} \\ &- g^{\circ}pq^{2} &&- \tfrac{3}{4}g'p^{2}q^{2} \\ &&&- (h^{\circ} - \tfrac{1}{3}{f^{\circ}}^{2})pq^{3} \end{alignedat}$

It seems worth while also to develop the area of the triangle $ABD$ into a series. For this development we may use the following conditional equation, which is easily derived from sufficiently obvious geometric considerations, and in which $S$  denotes the required area:

$$\frac{r\sin\psi}{n}. \frac{\partial S}{\partial p} + r\cos\psi. \frac{\partial S}{\partial q} = \frac{r\sin\psi}{n}. \int n\, dq$$

the integration beginning with $q = 0.$ From this equation we obtain, by the method of undetermined coefficients,

[7] $\displaystyle\begin{alignedat}{4} S = \tfrac{1}{2}pq &- \tfrac{1}{12}f^{\circ}p^{3}q &&- \tfrac{1}{20}f'p^{4}q &&- (\tfrac{1}{30}f'' - \tfrac{1}{60}{f^{\circ}}^{2})p^{5}q\quad\text{etc.} \\ &- \tfrac{1}{12}f^{\circ}pq^{3} &&- \tfrac{3}{40}g^{\circ}p^{3}q^{2} &&- \tfrac{1}{20}g'p^{4}q^{2} \\ &&&- \tfrac{7}{120}f'p^{2}q^{3} &&- (\tfrac{1}{15}h^{\circ} + \tfrac{2}{45}f'' + \tfrac{1}{60}{f^{\circ}}^{2})p^{3}q^{3} \\ &&&- \tfrac{1}{10}g^{\circ}pq^{4} &&- \tfrac{3}{40}g'p^{2}q^{4} \\ &&&&&- (\tfrac{1}{10}h^{\circ} - \tfrac{1}{30}{f^{\circ}}^{2})pq^{5} \end{alignedat}$