MATHEMATICS: WILSON AND MOORE 
273 
A GENERAL THEORY OF SURFACES 
By Edwin B. Wilson and C. L. E. Moore 
DEPARTMENT OF MATHEMATICS. MASSACHUSETTS INSTITUTE OF TECHNOLOGY 
Received by the Academy. March 23, 19 1 6 
Introduction. — ^Although the differential geometry of a ^-dimensional 
spread or variety Vk, embedded in an w-dimensional space Sn, has re- 
ceived considerable attention, in particular in the case k = n-1, the 
theory of surfaces F2 in the general Euclidean hyperspace Sn has been 
treated extensively by only three authors, Kommerell, Levi, and Segre,^ 
of whom the last was interested in projective properties, the first two in 
ordinary metric relations. The theory of V2 in Sn does, however, offer 
points of contact with elementary differential geometry which are fully 
as illuminating as those developed in the case of Vn-i in Sn- We have, 
therefore, undertaken to provide a general theory of surfaces which 
shall be independent of the number of dimensions of the containing 
space. 
The first thing to be determined in entering on such an extended study 
is the method of attack. The available methods were four: (1) The 
ordinary elementary method of starting with the finite equations of the 
surface and trying to generalize well-known geometric properties. This 
was followed by Kommerell. (2) The more advanced method of Levi, 
which depends upon the finite equations only for calculating certain in- 
variants / of rigid motion and further invariants (covariants) / of trans- 
formations of parameters upon the surface. (3) The vectorial method 
of Gibbs and others which bases the work upon the properties of the 
linear vector function expressing the relation between an infinitesimal 
displacement in the surface and the infinitesimal change in the unit 
normal (n-2) -space or the unit tangent plane. (4) The method of Ricci's 
absolute differential calculus, or Maschke's symbolic invariantive method, 
which develops the theory from the point of view of the fundamental 
quadratic differential forms defining the surface. 
We adopted the last method because of the possibility of sharply dif- 
ferentiating (1) those properties of surfaces which depend on the first 
fundamental form, and belong to the surface as any one of the infinite 
class of mutually appKcable surfaces, from (2) the properties which fol- 
low from the first and second fundamental forms taken together and 
which belong to the surface as a rigid surface. The adoption of Ricci^s 
rather than Maschke's form of analysis was further dictated by the fact 
that Ricci^ had developed the ordinary theory of surfaces consistently by 
his method, and all his work upon the first part (1), holding without 
