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 ${\textstyle \beta ,}$ ${\textstyle \gamma ,}$ which is done by multiplying them by ${\textstyle bc'-cb',}$ ${\textstyle b'C-c'B,}$ ${\textstyle cB-bC}$ 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 ${\textstyle \alpha ,}$ ${\textstyle \gamma }$ or ${\textstyle \alpha ,}$ ${\textstyle \beta }$ 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 ${\textstyle \alpha '',}$ ${\textstyle \beta '',}$ ${\textstyle \gamma ''}$ 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 ${\textstyle \alpha ',}$ ${\textstyle \beta ',}$ ${\textstyle \gamma '}$ 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

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