Transformations of Applicable Conjugate Nets of Curves on Surfaces

In a previous papert we developed the theory of transformations T of conjugate systems of curves on a surface into similar systems on other surfaces. In the present paper we are concerned with the application of these results to a particular class of conjugate systems, namely those which are applicable to one or more other systems. Thus if S and S are two surfaces upon which the corresponding conjugate system is parametric, we say that the parametric nets are applicable when corresponding first fundamental coefficients are equal. In order to give this definition analytic form, we suppose that the cartesian coordinates of S and S are x, y, z and x, I, z respectively. Since we are interested particularly in parametric nets, we designate them by N (x) and N (xt), or merely by N and N whenever it is not necessary to specify the coordinates. The first fundamental coefficients are defined by