1. When it becomes necessary in practical work to decide on a system of curves for describing frequency distributions, we have to bear in mind that

1. (1) Any expression used must be a graduation formula; it must remove the roughness of the material.

2. (2) The formula must not be too complicated. In particular there must not be too many constants in the formula. Also we wish to avoid use of high moments. The higher the moment the more liable it is to error when deduced from ungraduated observations; this is clear, when we remember that the ends of the experiences are multiplied by the highest numbers and their powers.

3. (3) There must be a systematic method of approaching frequency distributions.

2. Now, considering the more obvious characteristics of frequency distributions, we find they generally start at zero, rise to a maximum, and then fall sometimes at the same but often at a different rate. At the ends of the distribution there is often high contact. This means, mathematically, that a series of equations y = f(x), y = φ(x), etc., must be chosen, so that in each equation of the series dy/dx = 0 for certain values of x, namely, at the maximum and at the end of the curve where there is contact with the axis of x.

The above suggests that dy/dx may be put equal to [y×(x+a)]/F(x); then, if y = 0, dy/dx = 0, and there is, therefore, contact at one end, of the curve, while if x = −a, dy/dx = 0, and we have the maximum we require.