Page:General Investigations of Curved Surfaces, by Carl Friedrich Gauss, translated into English by Adam Miller Hiltebeitel and James Caddall Morehead.djvu/32

 $$\begin{aligned} C^{3}U &= -\alpha Aa'b' - \beta Ba'b' - \gamma Ca'b' \\ &\quad+ \alpha' A(ab' + ba') + \beta' B(ab' + ba') + \gamma' C(ab' + ba') \\ &\quad- \alpha Aab - \beta Bab - \gamma'' Cab \\ C^{3}V &= \alpha Aa'^{2} + \beta Ba'^{2} + \gamma Ca'^{2} \\ &\quad- 2\alpha' Aaa' - 2\beta' Baa' - 2\gamma' Caa' \\ &\quad+ \alpha Aa^{2} + \beta Ba^{2} + \gamma'' Ca^{2} \end{aligned}$$

Hence, if we put, for the sake of brevity,

$$\begin{alignedat}{4} &A\alpha  &&+ B\beta   &&+ C\gamma   &&= &D&\quad.\quad.\quad.\quad.\quad.\quad.\quad{(1)} \\ &A\alpha' &&+ B\beta'  &&+ C\gamma'  &&= &D'&\quad.\quad.\quad.\quad.\quad.\quad.\quad{(2)} \\ &A\alpha &&+ B\beta &&+ C\gamma &&= &D&\quad.\quad.\quad.\quad.\quad.\quad.\quad{(3)} \end{alignedat}$$

we shall have

$$\begin{aligned} C^{3}T &= Db'^{2} - 2D'bb' + D'' b^{2} \\ C^{3}U &= -Da'b' + D'(ab' + ba') - D''ab \\ C^{3}V &= Da'^{2} - 2D'aa' + D''a^{2} \end{aligned}$$

From this we find, after the reckoning has been carried out,

$$C^{6}(TV - U^{2}) = (DD - D'^{2}) (ab' - ba')^{2} = (DD - D'^{2}) C^{2}$$

and therefore the formula for the measure of curvature

$$k = \frac{DD'' - D'^{2}}{(A^{2} + B^{2} + C^{2})^{2}}$$

11.

By means of the formula just found we are going to establish another, which may be counted among the most productive theorems in the theory of curved surfaces. Let us introduce the following notation:

$$ \begin{aligned} &\begin{alignedat}{4} &a^{2} &&+ b^{2} &&+ c^{2} &&= E \\ &aa' &&+ bb' &&+ cc' &&= F \\ &a'^{2} &&+ b'^{2} &&+ c'^{2} &&= G \end{alignedat}\\ &\begin{alignedat}{7} &a &&\alpha &&+b &&\beta  &&+c &&\gamma  &&= m &\quad.\quad.\quad.\quad.\quad.\quad.\quad{(4)} \\ &a &&\alpha' &&+b &&\beta' &&+c &&\gamma' &&= m'&\quad.\quad.\quad.\quad.\quad.\quad.\quad{(5)} \\ &a &&\alpha&&+b &&\beta&&+c &&\gamma&&= m&\quad.\quad.\quad.\quad.\quad.\quad.\quad{(6)} \\ &a'\,&&\alpha &&+b'\,&&\beta  &&+c'\,&&\gamma  &&= n&\quad.\quad.\quad.\quad.\quad.\quad.\quad{(7)} \\ &a'&&\alpha' &&+b'&&\beta' &&+c'&&\gamma' &&= n'&\quad.\quad.\quad.\quad.\quad.\quad.\quad{(8)} \\ &a'&&\alpha&&+b'&&\beta&&+c'&&\gamma&&= n&\quad.\quad.\quad.\quad.\quad.\quad.\quad{(9)} \end{alignedat}\\ &A^{2} + B^{2} + C^{2} = EG - F^{2} = \Delta \end{aligned}$$