Page:General Investigations of Curved Surfaces, by Carl Friedrich Gauss, translated into English by Adam Miller Hiltebeitel and James Caddall Morehead.djvu/48

$${\begin{alignedat}{2}&{\sqrt {E}}.\sin(\psi -\omega )=m.{\frac {\partial \phi }{\partial p}}&\quad .\quad .\quad .\quad .\quad .\quad .\quad {(3)}\\&{\sqrt {G}}.\sin \psi =m.{\frac {\partial \phi }{\partial q}}\quad .\quad .&\quad .\quad .\quad .\quad .\quad .\quad .\quad {(4)}\end{alignedat}}$$

But the last and the next to the last equations of the preceding article give

$${\begin{aligned}EG-F^{2}=E\left({\frac {\partial r}{\partial q}}\right)^{2}-2F.{\frac {\partial r}{\partial p}}.{\frac {\partial r}{\partial q}}+G\left({\frac {\partial r}{\partial p}}\right)^{2}.&\quad .\quad {(5)}\;\quad \qquad \\\left(E.{\frac {\partial r}{\partial q}}-F.{\frac {\partial r}{\partial p}}\right).{\frac {\partial \phi }{\partial q}}=\left(F.{\frac {\partial r}{\partial q}}-G.{\frac {\partial r}{\partial p}}\right).{\frac {\partial \phi }{\partial p}}&\quad .\quad {(6)}\;\quad \qquad \end{aligned}}$$

From these equations must be determined the quantities ${\textstyle r,}$ ${\textstyle \phi ,}$ ${\textstyle \psi }$ and (if need be) ${\textstyle m,}$ as functions of ${\textstyle p}$ and ${\textstyle q.}$ Indeed, integration of equation (5) will give ${\textstyle r;}$ ${\textstyle r}$ being found, integration of equation (6) will give ${\textstyle \phi ;}$ and one or other of equations (1), (2) will give ${\textstyle \psi }$ itself. Finally, ${\textstyle m}$ is obtained from one or other of equations (3), (4).

The general integration of equations (5), (6) must necessarily introduce two arbitrary functions. We shall easily understand what their meaning is, if we remember that these equations are not limited to the case we are here considering, but are equally valid if ${\textstyle r}$ and ${\textstyle \phi }$ are taken in the more general sense of Art. 16, so that ${\textstyle r}$ is the length of the shortest line drawn normal to a fixed but arbitrary line, and ${\textstyle \phi }$ is an arbitrary function of the length of that part of the fixed line which is intercepted between any shortest line and an arbitrary fixed point. The general solution must embrace all this in a general way, and the arbitrary functions must go over into definite functions only when the arbitrary line and the arbitrary functions of its parts, which ${\textstyle \phi }$ must represent, are themselves defined. In our case an infinitely small circle may be taken, having its centre at the point from which the distances ${\textstyle r}$ are measured, and ${\textstyle \phi }$ will denote the parts themselves of this circle, divided by the radius. Whence it is easily seen that the equations (5), (6) are quite sufficient for our case, provided that the functions which they leave undefined satisfy the condition which ${\textstyle r}$ and ${\textstyle \phi }$ satisfy for the initial point and for points at an infinitely small distance from this point.

Moreover, in regard to the integration itself of the equations (5), (6), we know that it can be reduced to the integration of ordinary differential equations, which, however, often happen to be so complicated that there is little to be gained by the reduction. On the contrary, the development in series, which are abundantly sufficient for practical requirements, when only a finite portion of the surface is under consideration, presents no difficulty; and the formulæ thus derived open a fruitful source for