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 $R,$ 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:

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