The critical behaviour of the Blume Emery Griffiths (BEG) spin 1 Ising
model on a diamond lattice is studied within the framework of a finit
e cluster theory. An expansion technique for the cluster identities th
at allows the kinematic properties of the spin operators to be account
ed for in a systematic way is employed. The location of the tri-critic
al points in temperature versus single-ion anisotropy phase space is d
etermined for a range of biquadratic exchange interaction strengths.