In order that the final curve may be written in terms of the co-ordinates x and y the equation of the base or generating normal probability curve is written:
where t denotes abscissas and z ordinates.
Let the abscissas of the transformed curve be functions of the corresponding abscissas of the base curve. Then it may be assumed that x can be developed in powers of t, and hence we may write on omitting fourth and higher powers,
x = a(t + κt2 + λt3),
where a, κ and λ are constants to be determined in “fitting” the curve.
Since x denotes the value of a measurement and y the frequency of x, that is, the number of individuals possessing that value of x, the magnitude of an element of area denotes the number of individuals between two values of x. Obviously, therefore, if the transformation is to be of concrete value the magnitude of an element of area must not be altered, though of course the shape will be changed. Hence
y dx = z dt,
and