Let us consider two light rays of the same frequency in the $x$-$y$ plane forming the angles $\alpha$ and $\alpha+d\alpha$ with the axis $x$. The four-momenta of the corresponding photons are
$$p_1=\left (\frac{E}{c}, \frac{E}{c}\cos{\alpha}, \frac{E}{c}\sin{\alpha}
\right ),\;\;\;\mathrm{and}\;\;\;
p_2=\left (\frac{E}{c}, \frac{E}{c}\cos{(\alpha+d\alpha)},
\frac{E}{c}\sin{(\alpha+d\alpha)}\right ).$$
From the invariance of the scalar product
$$p_1\cdot p_2=\frac{E^2}{c^2}(1-\cos{(d\alpha)})\approx \frac{1}{2}
\frac{E^2}{c^2}(d\alpha)^2,$$
we get $$E\,d\alpha=E^\prime\,d\alpha^\prime, \tag 1$$ where the primed quantities are measured in the inertial frame $S^\prime$ moving with some velocity $V$ along the $x$-axis. On the other hand, transverse components of the 3-momentum is not changed under the Lorentz transformations and, therefore, $$\frac{E}{c}\sin{\alpha}=\frac{E^\prime}{c}\sin{\alpha^\prime}.\tag 2$$ Combining (1) and (2), we get
$$\frac{d\alpha}{\sin{\alpha}}=\frac{d\alpha^\prime}{\sin{\alpha^\prime}}. \tag 3$$ Hence this differential equation describes the (relativistic) aberration of light. An integration constant $n$ in its solution $$\cos{\alpha}=\frac{\cos{\alpha^\prime}+n}{1+n\,\cos{\alpha^\prime}}\tag 4$$ can be fixed in the following way. Let $\alpha^\prime=\pi/2$, then it is evident that the $x$-component of the photon's velocity $c_x=c\,\cos{\alpha}$ in the original "stationary" frame equals to the $S^\prime$ frames velocity $V$. on the other hand, from (4) in this case we have $\cos{\alpha}=n$. Therefore, $c\,n=V$ and $n=V/c$.

P.S. The relation with the spherical trigonometry (see William Chauvenet's book mentioned in the comments) can be established, I think, in light of Arnold Sommerfeld's this old idea: http://en.wikisource.org/wiki/Translation:On_the_Composition_of_Velocities_in_the_Theory_of_Relativity (On the Composition of Velocities in the Theory of Relativity). Of course, in fact, the real geometry of Einstein's velocity addition is the hyperbolic (Lobachevsky-Boljai) geometry - "the imaginary counter-image of the spherical geometry": http://en.wikisource.org/wiki/Translation:On_the_Non-Euclidean_Interpretation_of_the_Theory_of_Relativity (On the Non-Euclidean Interpretation of the Theory of Relativity, by Vladimir Varicak, 1912).

exactlythe one I had, and none is asserted to be related to Villarceau circles. I don't recall having come across the term "differential variation" before. $\endgroup$ – Michael Hardy May 3 '14 at 2:013more comments