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Bolza continues and describes Gauss derivation of the following formula (i didn't include the derivation):

$$K_gds - d\theta = Pdp + Qdq$$

where $P,Q$ are the following functions:

$$P = \frac{EF_p - \frac{1}{2}EE_q - \frac{1}{2}FE_p}{E\sqrt{EG-F^2}},Q = \frac{\frac{1}{2}EG_p - \frac{1}{2}FE_q}{E\sqrt{EG-F^2}} $$

Bolza continues and describes Gauss's derivation of the following formula (I didn't include the derivation):

$$(*) K_gds - d\theta = Pdp + Qdq$$

where $ds$ is an element of length of a curve, $\theta$ is the angle between the positive p-direction and the tangent to the curve, and $P,Q$ are the following functions:

$$P = \frac{EF_p - \frac{1}{2}EE_q - \frac{1}{2}FE_p}{E\sqrt{EG-F^2}},Q = \frac{\frac{1}{2}EG_p - \frac{1}{2}FE_q}{E\sqrt{EG-F^2}} $$

(Taking a line integral of both sides of equation (*) along a closed loop and applying Green's theorem on the right side enables to write the right side in the form:$\int\int(\frac {\partial P}{\partial q}-\frac{\partial Q}{\partial p})dpdq$. Apparently the term $\frac{\partial P}{\partial q}-\frac{\partial Q}{\partial p}$ is none other than the Gaussian curvature $K$.)

Although i've already accepted one answer, i added this answer in order to clarify what is known about Gauss's work towards the general Gauss-Bonnet theorem and what is a matter of speculation; this distinction is not clear in Mark Yasuda's answer, and one might get from it incorrect impression about the roots of Gauss differential geometric ideas.

As an illustration of several ideas expressed in this answer, i've added this figure.

Although I've already accepted one answer, I added this answer in order to clarify what is known about Gauss's work towards the general Gauss-Bonnet theorem and what is a matter of speculation; this distinction is not clear in Mark Yasuda's answer, and one might get from it incorrect impression about the roots of Gauss differential geometric ideas.

As an illustration of several ideas expressed in this answer, I've added this figure.

My indications are based upon reading Oscar Bolza's treatise on Gauss's contibution to the calculus of variations and differential geometry; a section of it is devoted to "the geodetic curvature and the theorem of total curvature". In this section, Bolza analyzes Gauss's unpublshed manuscript entitled "Die Sietenkrummung" ("the side curvature"), which dates back to the years before his publication on differential geometry (the years 1822-1825). Gauss introduces in this manuscript the "extrinsic" definition of the geodesic curvature $K_g$, and gives an "intrinsic" formula for it in term of the coefficients of the first fundamental form $E,F,G$ and their first and second derivatives with respect to the curvilinear coordinates $p,q$ - this formula proves that the geodesic curvature is an isometric invariant of the surface, in contrast to the apparently extrinsic definition of it (like in the case of the "Theorema Egregium" on Gauss curvature). This result was rediscovered by Ferdinand Minding about a decade later.

My indications are based upon reading Oscar Bolza's treatise on Gauss's contibution to the calculus of variations and differential geometry; a section of it is devoted to "the geodetic curvature and the theorem of total curvature". In this section, Bolza analyzes Gauss's unpublshed manuscript entitled "Die Sietenkrummung" ("the side curvature"), which dates back to the years before his publication on differential geometry (the years 1822-1825). Gauss introduces in this manuscript the "extrinsic" definition of the geodesic curvature $K_g$ at a given point of a curve embedded in the surface , and gives an "intrinsic" formula for it in term of the coefficients of the first fundamental form $E,F,G$ and their first and second derivatives with respect to the curvilinear coordinates $p,q$ - this formula proves that the geodesic curvature is an isometric invariant of the surface, in contrast to the apparently extrinsic definition of it (like in the case of the "Theorema Egregium" on Gauss curvature). This result was rediscovered by Ferdinand Minding about a decade later.