Representations Between Surfaces Defined by Means of the Quadratic Form Which Determines Their Modulus of Deformation

In earlier works I have had occasion to study representations between surfaces, in which it is assumed that, on one of the surfaces (which we shall call the object surface) there is arbitrarily assigned the quadratic form as a function of position or the surface, which fixes in every direction the modulus of linear deformation of the representation itself. If the quadratic form is quite general, limited only in that it should be positive definite, and if its coefficients are continuous functions together with their first and second derivatives, the representation will be said to be affine, since it establishes an affinity between the neighbourhood of the first order of the object surface ∑, and the corresponding neighbourhood of the second surface \( \bar \Sigma \) (which we shall call the image surface).