**Abstract** : The magnetoresistance of Ni 81 Fe 19 and Co 83 Gd 17 ferromagnetic thin films is measured in Corbino disk geometry, and compared to the magnetoresistance of the same films measured in the Hall-bar geometry. The symmetry of the magnetoresistance profiles is drastically modified by changing the geometry of the sample, i.e., by changing the boundary conditions. These properties are explained in a simple model, showing that the Corbino magnetoresistance is defined by the potentiostatic boundary conditions while the Hall-bar magnetoresistance is defined by galvanostatic boundary conditions. The Hall effect was first measured in 1879 by Hall [1] by applying a magnetic field to a conducting slab contacted to an electric generator at the extremities. Later on, Corbino [2] found a similar effect by applying a magnetic field on a disk geometry with two concentric electrodes. Quickly the question arose on whether the effect measured by Corbino (the so-called Corbino effect) in a disk and by Hall in a bar have the same origin. In 1914, Adams and Chapman measured the Corbino effect in many different metals [3] by using an oscillating current flowing from the center of the disk to its outer. Adams concluded in 1915 that "the Corbino effect is, essentially, the same as the Hall effect" [4]. However, the question remains about the exact meaning of the adverb "essentially." In the 1950's, the Hall effect in the Corbino geometry was studied for its practical applications. The magnetoresistance of InSb slabs was shown to depend strongly on the shape of the samples [5]. The reason is that near the current injection edge, the Hall electric field is shorted and a transverse electric current appears which causes an increase of the resistance as in the Corbino geometry [6-9]. Accordingly, one can see the Corbino geometry as the extreme scenario where the Hall electric field is zero everywhere and a Hall current is flowing, or, in other terms, one can view the Corbino disk as a Hall bar in which the electrostatic charge accumulation is reduced to zero everywhere. The system cannot generate a voltage between the edges so that a Hall current is flowing and the Joule heating is higher than in the Hall bar for an equivalent volume [10-12]. The mechanism responsible for both the Hall effect and the Corbino effect is indeed the same, but the Corbino disk is a device that is more constrained than the Hall bar, due to the change of the boundary conditions. At the turn of the last century, the emergence of spintronics has shown the possibility of exploiting spin-polarized currents * jean-eric.wegrowe@polytechnique.edu and spin-dependent potentials, and has paved the way to the realization of new electronic devices. Recently, various developments about the spin-Hall effects (anomalous Hall effect, spin-Hall effect, spin-pumping effect, spin-Seebeck effects, etc. [13,14]) tend to show that the usual Hall-bar conditions with spin relaxation could be turned into "Corbino-like boundary conditions," in the sense that the electric charge accumulation drops to zero at the edges and a pure spin current can be generated instead of a Hall voltage [11]. In this context, the goal of this Rapid Communication is to study NiFe and GdCo ferromagnetic Corbino disks and Hall bars, in order to understand the behavior of the magnetoresistance [13,15-17] when the boundary conditions are switched (by changing the geometry) from spin current to spin-dependent voltage. The alloys Ni 81 Fe 19 and Co 83 Gd 17 are chosen for their maximum contribution to the anisotropic magnetoresistance and the anomalous Hall magnetoresistance (that defines the anomalous Hall angle), respectively. First, we will present our measures of Corbino magne-toresistance performed on NiFe and CoGd rings. The results are then analyzed in the framework of the generalized Ohm's law by defining the Corbino magnetotransport coefficients C as a function of the usual Hall-bar coefficients [see Eq. (12) below]. The consistency of the proposed explanation is checked independently, by measuring the magnetotransport coefficients of the Hall bar. The samples studied are 20-nm-thick layers of Ni 81 Fe 19 and Co 83 Gd 17 sputtered on glass substrates. The magnetic layers are sandwiched between 5-nm-thick Ta buffers and 3-nm-thick Pt caps. The magnetic properties of the thin layers have been previously studied [18] (see Supplemental Material [19]). The sample magnetization is uniform for quasistatic states, although nonuniform states could take place at low magnetic fields (this regime is, however, not considered in the present study). The NiFe is textured with small uniaxial anisotropy lying in the sample plane. The coercivity field in the in-plane geometry is of the order of 1 mT. The out-2469-9950/2018/98(22)/220405(5) 220405-1