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 ${\textstyle dz}$ 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 ${\textstyle k}$ the following symmetric expression:

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

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

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