Using a continuum theory which allows for changes in variables which represent the phase and biaxiality of the liquid crystal as well as the director field, the core structure of plus or minus one half and plus or minus one strength disclination lines is investigated. Under certain approximations analytical solutions are found near to the centre of the disclination. Good agreement is found with numerical solutions for the full problem. Using a continuation package (AUTO), the changes to these numerical solutions are then considered as various parameters are altered. The model exhibits a first-order phase transition near to the clearing point temperature induced by the presence of the disclination core. If the disclination was not present, this phase transition would occur when the liquid crystalline state loses stability at a higher temperature.