Page:General Investigations of Curved Surfaces, by Carl Friedrich Gauss, translated into English by Adam Miller Hiltebeitel and James Caddall Morehead.djvu/33

 Let us eliminate from the equations 1, 4, 7 the quantities $\beta,$  $\gamma,$ which is done by multiplying them by $bc' - cb',$  $b'C - c'B,$  $cB - bC$  respectively and adding. In this way we obtain

$$\begin{array}{c} \mathbf{(}A(bc' - cb') + a(b'C - c'B) + a'(cB - bC)\mathbf{)}\alpha \\ = D(bc' - cb') + m(b'C - c'B) + n(cB - bC) \end{array}$$

an equation which is easily transformed into

$$AD = \alpha\Delta + a(nF - mG) + a'(mF - nE)$$

Likewise the elimination of $\alpha,$  $\gamma$ or $\alpha,$  $\beta$  from the same equations gives

$$\begin{alignedat}{4} &BD &&= \beta \Delta &&+ b(nF - mG) &&+ b'(mF - nE) \\ &CD &&= \gamma\Delta &&+ c(nF - mG) &&+ c'(mF - nE) \end{alignedat}$$

Multiplying these three equations by $\alpha,$  $\beta,$  $\gamma''$ respectively and adding, we obtain

$$DD = (\alpha\alpha+ \beta\beta + \gamma\gamma)\Delta + m(nF - mG) + n(mF - nE) \quad.\quad.\quad.{(10)}$$

If we treat the equations 2, 5, 8 in the same way, we obtain

$$\begin{alignedat}{4} {4} &AD' &&= \alpha'\Delta &&+ a (n'F - m'G) &&+ a'(m'F - n'E) \\ &BD' &&= \beta' \Delta &&+ b (n'F - m'G) &&+ b'(m'F - n'E) \\ &CD' &&= \gamma'\Delta &&+ c (n'F - m'G) &&+ c'(m'F - n'E) \end{alignedat}$$

and after these equations are multiplied by $\alpha',$  $\beta',$  $\gamma'$ respectively, addition gives

$$D'^{2} = (\alpha'^{2} + \beta'^{2} + \gamma'^{2})\Delta + m'(n'F - m'G) + n'(m'F - n'E)$$

A combination of this equation with equation (10) gives

$$\begin{alignedat}{1} DD - D'^{2} = (&\alpha\alpha + \beta\beta + \gamma\gamma - \alpha'^{2} - \beta'^{2} - \gamma'^{2})\Delta \\ &+ E(n'^{2} - nn) + F(nm - 2m'n' + mn) + G(m'^{2} - mm) \end{alignedat}$$

It is clear that we have

$$ \frac{\partial E}{\partial p} = 2m,\; \frac{\partial E}{\partial q} = 2m',\; \frac{\partial F}{\partial p} = m' + n,\; \frac{\partial F}{\partial q} = m'' + n',\; \frac{\partial G}{\partial p} = 2n',\; \frac{\partial G}{\partial q} = 2n'',$$

or

$$\begin{aligned} m  &= \tfrac{1}{2}\, \frac{\partial E}{\partial p}, & m' &= \tfrac{1}{2}\, \frac{\partial E}{\partial q}, & m'' &= \frac{\partial F}{\partial q} - \tfrac{1}{2}\, \frac{\partial G}{\partial p} \\ n  &= \frac{\partial F}{\partial p} - \tfrac{1}{2}\, \frac{\partial E}{\partial q}, & n' &= \tfrac{1}{2}\, \frac{\partial G}{\partial p}, & n'' &= \tfrac{1}{2}\, \frac{\partial G}{\partial q} \end{aligned}$$

Moreover, it is easily shown that we shall have

$$\begin{array}{c}\displaystyle \alpha\alpha + \beta\beta + \gamma\gamma'' - \alpha'^{2} - \beta'^{2} - \gamma'^{2} = \frac{\partial n}{\partial q} - \frac{\partial n'}{\partial p}  = \frac{\partial m''}{\partial p} - \frac{\partial m'}{\partial q} \\\displaystyle = -\tfrac{1}{2} \cdot \frac{\partial^{2}E}{\partial q^{2}} + \frac{\partial^{2}F}{\partial p. \partial q} - \tfrac{1}{2} \cdot \frac{\partial^{2}G}{\partial p^{2}} \end{array}$$