Tables of Functions of First and Second Partial Derivatives of Legendre Polynomials includes an introduction to the project initiated by the University of Michigan's Department of Chemical Engineering for determining droplet sizes and distribution in atomized sprays using Mie theory. The need for extended tables of light-scattering functions arose, leading to computations made by IBM Calculating Punch and ENIAC. The publication contains mathematical definitions of the tabulated functions, an explanation of the tabulated functions, the method of computation, the accuracy of the tables, acknowledgments, references, and the tables themselves. The accuracy of the tables has been ensured by calculating to seven decimal places and duplicating every calculation to check for errors.