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We present calculations related to the incorporation of the presence of a third star companion of the eclipsing binary star 44i Boötes into the analysis of the period variations of the eclipsing pair. We have used Mathematica to calculate the effect of the third star according to five different sets of orbital parameters and we have found a best fit to the data. We give a new ephemeris to predict future eclipse times that incorporates a steady period increase of 8.53 micro seconds/Epoch, the so called Light-Time effect that takes the third star into consideration, and also a sinusoidal term with a period of 61.5 years. We found that adding the Light-Time effect decreases the already known rate of increase of period of the eclipsing system. This consideration of the third star also leaves intact sinusoidal residuals, implying that these are in fact a true phenomena of the system.