On the curvature of surfaces

Abstract

The principal object of this paper is, to remove the obscurity in which that part of the theory of the curvature of surfaces which relates to umbilical points has been left by Monge and Dupin, to whom, however, subsequently to the labours of Euler, we are chiefly indebted for a comprehensive and systematic theory of the curvature of surfaces. In it the author shows, that the lines of curvature at an umbilic are not, as at other points on a surface, two in number, or, as had been stated by Dupin, limited; but that they proceed in every possible direction from the umbilic. The obscurity complained of is attributed to the inaccurate conceptions entertained by Monge and Dupin, of the import of the symbol 0/0 in the analytical discussion of this question, the equation which determines the directions of the lines of curvature taking the form 0( dy / dx ) 2 + 0( dy / dx ) + 0 = 0 at an umbilic. After stating that Dupin has been guided by the determination of the differential calculus, the author remarks, that in no case is the differential calculus competent to decide whether 0/0, the form which a general analytical result takes in certain particular hypotheses, as to the arbitrary quantities entering that result, has or has not innumerable values. He then states the principle, that those values of the arbitrary quantities (and none else) which render the equations of condition indeterminate must also render the final result, to which they lead, equally indeterminate; and that, therefore, when such result assumes the form 0/0, its true character is to be tested by the equations that have led to it, after these have been modified by the hypothesis from which that form has arisen.