In 1909,
Augustus Edward Hough Love introduced the values h and k which characterize the overall
elastic response of the Earth to the tides ― Earth tides or body tides.^{
[1]} Later, in 1912, Toshi Shida added a third Love number, l, which was needed to obtain a complete overall description of the solid Earth's response to the
tides.^{
[2]}

Definitions

The Love number h is defined as the ratio of the body tide to the height of the static
equilibrium tide;^{
[3]} also defined as the vertical (radial) displacement or variation of the planet's elastic properties. In terms of the tide generating potential $V(\theta ,\phi )/g$, the displacement is $hV(\theta ,\phi )/g$ where $\theta$ is latitude, $\phi$ is east longitude and $g$ is acceleration due to gravity.^{
[4]} For a hypothetical solid Earth $h=0$. For a liquid Earth, one would expect $h=1$. However, the deformation of the sphere causes the potential field to change, and thereby deform the sphere even more. The theoretical maximum is $h=2.5$. For the real Earth, $h$ lies between 0 and 1.

The Love number k is defined as the cubical dilation or the ratio of the additional potential (self-reactive force) produced by the deformation of the deforming potential. It can be represented as $kV(\theta ,\phi )/g$, where $k=0$ for a rigid body.^{
[4]}

The Love number l represents the ratio of the horizontal (transverse) displacement of an element of mass of the planet's crust to that of the corresponding static ocean tide.^{
[3]} In potential notation the transverse displacement is $l\nabla (V(\theta ,\phi ))/g$, where $\nabla$ is the horizontal
gradient operator. As with h and k, $l=0$ for a rigid body.^{
[4]}

Values

According to Cartwright, "An elastic solid spheroid will yield to an external tide potential $U_{2}$ of
spherical harmonic degree 2 by a surface tide $h_{2}U_{2}/g$ and the self-attraction of this tide will increase the external potential by $k_{2}U_{2}$."^{
[5]} The magnitudes of the Love numbers depend on the rigidity and mass distribution of the spheroid. Love numbers $h_{n}$, $k_{n}$, and $l_{n}$ can also be calculated for higher orders of spherical harmonics.

For elastic Earth the Love numbers lie in the range: $0.616\leq h_{2}\leq 0.624$, $0.304\leq k_{2}\leq 0.312$ and $0.084\leq l_{2}\leq 0.088$.^{
[3]}

For Earth's tides one can calculate the tilt factor as $1+k-h$ and the gravimetric factor as $1+h-(3/2)k$, where subscript two is assumed.^{
[5]}

^TOSHI SHIDA, On the Body Tides of the Earth, A Proposal for the International Geodetic Association, Proceedings of the Tokyo Mathematico-Physical Society. 2nd Series, 1911-1912, Volume 6, Issue 16, Pages 242-258, ISSN 2185-2693,
doi:
10.11429/ptmps1907.6.16_242.

^
^{a}^{b}^{c}"Tidal Deformation of the Solid Earth: A Finite Difference Discretization", S.K.Poulsen; Niels Bohr Institute, University of Copenhagen; p 24;
[1]Archived 2016-10-11 at the
Wayback Machine

^
^{a}^{b}^{c}Earth Tides; D.C.Agnew, University of California; 2007; 174

^
^{a}^{b}Tides: A Scientific History; David E. Cartwright; Cambridge University Press, 1999,
ISBN0-521-62145-3; pp 140–141,224