# Dictionary:Euler’s homogeneity equation

${\displaystyle (x-x_{0}){\frac {dF}{dx}}+(y-y_{0}){\frac {dF}{dy}}+(z-z_{0}){\frac {dF}{dz}}=N(B-F)}$,
where ${\displaystyle (x_{0},y_{0},z_{0})}$ is the source location whose magnetic field ${\displaystyle F}$ is measured at ${\displaystyle (x,y,z)}$; B is the regional value of the total field; and ${\displaystyle N}$ is Euler’s structural index. ${\displaystyle N}$ is a measure of the rate of field change with distance. For example, the magnetic field of a sphere falls off as the cube ${\displaystyle (N=3)}$, of a pipe as the square ${\displaystyle (N=2)}$, of a thin dike linearly ${\displaystyle (N=1)}$, for a more or less linear basement fault or dyke ${\displaystyle (N=0.5)}$, of a semi-infinite body, not at all ${\displaystyle (N=0)}$. An Euler depth estimate increases with increased ${\displaystyle N}$. Real bodies are simulated by a superposition of bodies.