Page:General Investigations of Curved Surfaces, by Carl Friedrich Gauss, translated into English by Adam Miller Hiltebeitel and James Caddall Morehead.djvu/98

and therefore

$$\tan \phi =-{\frac {p}{q}}.$$

We have also

$$\cos \phi .dp+\sin \phi .dq-(p\sin \phi -q\cos \phi )\,d\phi =0.$$

If, therefore, we set, according to a well known theorem,

$${\begin{aligned}dp&=P\,dx+Q\,dy,\\dq&=Q\,dx+R\,dy,\end{aligned}}$$

then we have

$$(P\cos ^{2}\phi +2Q\cos \phi \sin \phi +R\sin ^{2}\phi )\,ds=(p\sin \phi -q\cos \phi )\,d\phi ,$$

therefore

$${\frac {1}{r}}={\frac {P\cos ^{2}\phi +2Q\cos \phi \sin \phi +R\sin ^{2}\phi }{p\sin \phi -q\cos \phi }},$$

or, since

$$\cos \phi ={\frac {\mp q}{\sqrt {p^{2}+q^{2}}}},\qquad \sin \phi ={\frac {\pm p}{\sqrt {p^{2}+q^{2}}}};$$

$$\pm {\frac {1}{r}}={\frac {Pq^{2}-2Qpq+Rp^{2}}{(p^{2}+q^{2})^{3/2}}}.$$

4.

The ambiguous sign in the last formula might at first seem out of place, but upon closer consideration it is found to be quite in order. In fact, since this expression depends simply upon the partial differentials of ${\textstyle V,}$ and since the function ${\textstyle V}$ itself merely defines the nature of the curve without at the same time fixing the sense in which it is supposed to be described, the question, whether the curve is convex toward the right or left, must remain undetermined until the sense is determined by some other means. The case is similar in the determination of ${\textstyle \phi }$ by means of the tangent, to single values of which correspond two angles differing by ${\textstyle 180^{\circ }.}$ The sense in which the curve is described can be specified in the following different ways.

I. By means of the sign of the change in ${\textstyle x.}$ If ${\textstyle x}$ increases, then ${\textstyle \cos \phi }$ must be positive. Hence the upper signs will hold if ${\textstyle q}$ has a negative value, and the lower signs if ${\textstyle q}$ has a positive value. When ${\textstyle x}$ decreases, the contrary is true.

II. By means of the sign of the change in ${\textstyle y.}$ If ${\textstyle y}$ increases, the upper signs must be taken when ${\textstyle p}$ is positive, the lower when ${\textstyle p}$ is negative. The contrary is true when ${\textstyle y}$ decreases.

III. By means of the sign of the value which the function ${\textstyle V}$ takes for points not on the curve. Let ${\textstyle \delta x,}$ ${\textstyle \delta y}$ be the variations of ${\textstyle x,}$ ${\textstyle y}$ when we go out from the