  We compute rational $T$-equivariant elliptic cohomology of CP(V), where $T$
is the circle group, and CP(V) is the $T$-space of complex lines for a finite
dimensional complex $T$-representation V. Starting from an elliptic curve C
over the complex numbers and a coordinate data around the identity, we achieve
this computation by proving that the $T$-equivariant elliptic cohomology theory
$EC_T$ built in [Gre05], and the $T^2$-equivariant elliptic cohomology theory
$EC_{T^2}$ built in [Bar22] are 1x$T$-split. This result allows us to reduce
the computation of $EC_T(CP(V))$ to the computation of $T^2$-elliptic
cohomology of spheres of complex representations, already performed in [Bar22].
