When each infinitely small triangle on the one surface is depicted in an infinitely small and similar triangle on the other surface, the de piction is said to be conformal. It follows that corresponding angles on A and B are equal; also that ds'= kde, where ds is arc length on A measured from a point P, ds, the corresponding arc length on B and k a quantity depending on the position of P and independent of the direction of ds. The analytical side of con formal representation is completely resolved by recourse to thermal parameters. The arc element of A in the thermal parameters u, v is di` = a(du' + de); similarly, arc element of B in the thermal parameters u,, v, is de dv,'). The relations is, + iv= f (u + iv), — fi(u — iv), where f and f, are arbitrary conjugate functions establish a conformal representation, since, by virtue of these relations, ds'=kde. Any two surfaces can, in general, be conformally repre sented upon each other in an infinity of ways. The functions f and f, can be chosen to furnish the most advantageous conformal representa tion.
Two surfaces are applicable or developable upon each other if the corresponding infinitely small triangles are equal in all respects. This requires that corresponding arc elements shall be everywhere equal, namely, that in and v, shall be such functions of u and v as to trans form the first member of the equation 2F,du,dv, + = Ede + 2Fdu dv + Gdv' into the second member. The letters with sub scripts indicate the elements of the second sur face. In general, this transformation cannot be made, and hence two arbitrarily given sur faces are, in general, not developed upon each other. It is obvious that all surfaces derived from a given surface by bending without stretching (see 8) are applicable upon each other. Hence the parameters of any one of them are expressable in the parameters of the original surface. All the surfaces may, accord ingly, be assumed definite in the same parame ters u, v, whence it follows that the fundamen tal magnitudes E, F, G will be identically the same for the entire series of surfaces. The
three magnitudes E, F, G and all functions formed from them and their partial derivatives are invariants of bending. Some important conclusions can immediately be drawn from these statements. We observe that the left hand member of equation ( y ) is the Gauss measure of curvature and that the right-hand member is a function of E, F, G alone. We conclude that the Gauss curvature does not change in any deformation of a surface by bending. One notes also that equation (p ) depends only on E, F, G, whence the theorem that a geodesic curve remains a geodesic in the deformation by bending.
As earth-dwellers the most interesting depic tion to us is that of a sphere upon a plane. The sphere is not developable upon a plane and, therefore, any depiction is bound to be a distorted image of the original. A conformal representa tion will at least preserve angles, and the pic ture and original will be similar in the cor responding infinitely smallparts. The two best known examples of a conformal representation are the stereographic projection (Hipparchus, Ptolemy) and the projection of Mercator. Ex pressing the sphere of radius one in thermal parameters ui, 2u, 2v, 1 x ,Y 1 + 1 and a plane in thermal parameters u, v: x = u, y'=.v ; the stereographic projection is furnished by the relations u + iv = u — iv = or simply u = v =th. For the Mercator correspondence one sets up the relations • u, or ui = ea cos v, • = sin v. In the stereographic projection the circles of the sphere are represented by circles (or straight lines) in the plane ; in the Mercator map the meridians and parallels of latitude appear in the plane as a system of orthogonally intersecting right lines.