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 ${\textstyle \psi .}$ In like manner, for the calculation of the angle ${\textstyle \phi ,}$ series for ${\textstyle r\cos \phi }$ and ${\textstyle r\sin \phi }$ 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 ${\textstyle r\cos \phi ,}$ ${\textstyle r\sin \phi }$ are easily developed, whose first terms must evidently be ${\textstyle p,}$ ${\textstyle q}$ 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]

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

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