Interpolation is one of the most frequent processes in calculation, and yet it is the process in which most computers find the ordinary methods least satisfactory and most troublesome. Indeed, whenever linear interpolation is not practicable, it is usually worth while to find out a method depending on the nature the functions involved in the calculation, and use it in preference to the ordinary difference or Lagrangian formulae. In interpolation by differences there is the want of adequate tables of the coefficients, and worse than that, the necessity for watching the signs and the decimal points, a necessity which in these days calculating machines is relatively a great trouble. There is usually, moreover, a lack of system about interpolation by differences that makes it peculiarly susceptible to slips of working. In this connection I might mention a useful and not too well-known arrangement of the work for Newton's formula which Legendre gives in his Traité des fonctions elliptiqueg.