Page:General Investigations of Curved Surfaces, by Carl Friedrich Gauss, translated into English by Adam Miller Hiltebeitel and James Caddall Morehead.djvu/30

 or, eliminating $dz$ by means of the equation

$$\begin{array}{c} P\, dx + Q\, dy + R\, dz = 0, \\ R^{3}\, dt = (-R^{2}P' + 2PRQ - P^{2}R')\, dx + (PRP + QRQ - PQR' - R^{2}R)\, dy. \end{array}$$

In like manner we obtain

$$R^{3}\, du = (PRP + QRQ - PQR' - R^{2}R)\, dx + (-R^{2}Q' + 2QRP - Q^{2}R')\, dy$$

From this we conclude that

$$\begin{aligned} R^{3}T &= -R^{2}P' + 2PRQ'' - P^{2}R' \\ R^{3}U &= PRP + QRQ - PQR' - R^{2}R'' \\ R^{3}V &= -R^{2}Q' + 2QRP'' - Q^{2}R' \end{aligned}$$

Substituting these values in the formula of Art. 7, we obtain for the measure of curvature $k$ the following symmetric expression:

$$\begin{array}{c} (P^{2} + Q^{2} + R^{2})^{2}k = P^{2}(Q'R' - P''^{2}) + Q^{2}(P'R' - Q''^{2}) + R^{2}(P'Q' - R''^{2}) \\ + 2QR(QR - P'P'') + 2PR(PR - Q'Q'') + 2PQ(PQ - R'R'') \end{array}$$

10.

We obtain a still more complicated formula, indeed, one involving fifteen elements, if we follow the second general method of defining the nature of a curved surface. It is, however, very important that we develop this formula also. Retaining the notations of Art. 4, let us put also

$$\begin{aligned} \frac{\partial^{2}x}{\partial p^{2}} &= \alpha, & \frac{\partial^{2}x}{\partial p. \partial q} &= \alpha', & \frac{\partial^{2}x}{\partial q^{2}} &= \alpha'' \\ \frac{\partial^{2}y}{\partial p^{2}} &= \beta, & \frac{\partial^{2}y}{\partial p. \partial q} &= \beta', & \frac{\partial^{2}y}{\partial q^{2}} &= \beta'' \\ \frac{\partial^{2}z}{\partial p^{2}} &= \gamma, & \frac{\partial^{2}z}{\partial p. \partial q} &= \gamma', & \frac{\partial^{2}z}{\partial q^{2}} &= \gamma'' \\ \end{aligned}$$

and let us put, for brevity,

$$\begin{aligned} bc' - cb' &= A \\ ca' - ac' &= B \\ ab' - ba' &= C \end{aligned}$$

First we see that

$$A\, dx + B\, dy + C\, dz = 0,$$

or

$$dz = -\frac{A}{C}\, dx - \frac{B}{C}\, dy.$$