Page:General Investigations of Curved Surfaces, by Carl Friedrich Gauss, translated into English by Adam Miller Hiltebeitel and James Caddall Morehead.djvu/56

27.

If the curved surface is a sphere of radius ${\textstyle R,}$ we shall have

$$\alpha =\beta =\gamma =-2f^{\circ }={\frac {1}{R^{2}}};\quad f''=0,\quad g'=0,\quad 6h^{\circ }-{f^{\circ }}^{2}=0,$$

or

$$h^{\circ }={\frac {1}{24R^{4}}}.$$

Consequently, formula [14] becomes

$$A+B+C=\pi +{\frac {\sigma }{R^{2}}},$$

which is absolutely exact. But formulæ [11], [12], [13] give

$${\begin{aligned}A^{*}&=A-{\frac {\sigma }{3R^{2}}}-{\frac {\sigma }{180R^{4}}}(2p^{2}-q^{2}+4qq'-q'^{2})\\B^{*}&=B-{\frac {\sigma }{3R^{2}}}+{\frac {\sigma }{180R^{4}}}(p^{2}-2q^{2}+2qq'+q'^{2})\\C^{*}&=C-{\frac {\sigma }{3R^{2}}}+{\frac {\sigma }{180R^{4}}}(p^{2}+q^{2}+2qq'-2q'^{2})\end{aligned}}$$

or, with equal exactness,

$${\begin{aligned}A^{*}&=A-{\frac {\sigma }{3R^{2}}}-{\frac {\sigma }{180R^{4}}}(b^{2}+c^{2}-2a^{2})\\B^{*}&=B-{\frac {\sigma }{3R^{2}}}-{\frac {\sigma }{180R^{4}}}(a^{2}+c^{2}-2b^{2})\\C^{*}&=C-{\frac {\sigma }{3R^{2}}}-{\frac {\sigma }{180R^{4}}}(a^{2}+b^{2}-2c^{2})\end{aligned}}$$

Neglecting quantities of the fourth order, we obtain from the above the well-known theorem first established by the illustrious Legendre.

28.

Our general formulæ, if we neglect terms of the fourth order, become extremely simple, namely:

$${\begin{aligned}A^{*}&=A-{\tfrac {1}{12}}\sigma (2\alpha +\beta +\gamma )\\B^{*}&=B-{\tfrac {1}{12}}\sigma (\alpha +2\beta +\gamma )\\C^{*}&=C-{\tfrac {1}{12}}\sigma (\alpha +\beta +2\gamma )\end{aligned}}$$