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 ${\textstyle \psi +\phi }$ 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 ${\textstyle \psi +\phi ,}$ if we observe that its first term must be ${\textstyle {\frac {1}{2}}\pi ,}$ the radius being taken equal to unity and ${\textstyle 2\pi }$ denoting the circumference of the circle,

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

It seems worth while also to develop the area of the triangle ${\textstyle ABD}$ 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 ${\textstyle S}$ 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 ${\textstyle q=0.}$ From this equation we obtain, by the method of undetermined coefficients,

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