  The multiple images of lensed quasars provide evidence on the mass
distribution of the lensing galaxy. The lensing invariants are constructed from
the positions of the images, their parities and their fluxes. They depend only
on the structure of the lensing potential. The simplest is the magnification
invariant, which is the sum of the signed magnifications of the images. Higher
order invariants are the sums of products of the signed magnifications with
positive or negative powers of the position coordinates of the images. We
consider the case of the four and five image systems produced by elliptical
power-law galaxies. This paper provides simple contour integrals for evaluating
all their lensing invariants. For practical evaluation, this offers
considerable advantages over the algebraic methods used previously. To
illustrate the power of the contour integral method, we calculate full
expansions in the source offset for all lensing invariants in the presence of
arbitrary external shear. As an example, we use the elliptical power-law
galaxies to fit to the data on the four images of the Einstein Cross
(G2237+030). The lensing invariants play a role by reducing the dimensionality
of the parameter space in which the chi-squared minimisation proceeds with
consequent gains in accuracy and speed.
