animal and a powerful Leyden battery, it was concluded that the quantity of force in each shock of the former was very great. It was also ascertained by all the tests capable of bearing on the point, that the current of electricity was, in ever}' case, from the anterior parts of the animal through the water or surrounding conductors to the posterior parts. The author then proceeds to express his hope that by means of these organs and the similar parts of the Torpedo, a relation as to action and re-action of the electric and nervous powers may be established experimentally ; and he briefly describes the form of experiment which seems likely to yield positive results of this kind.
December 20, 1838.
JOHN GEORGE CHILDREN, Esq., V.P., in the Chair.
Prof. Louis Agassiz, and Prof. Carl. Fred. Philip von Martins, were severally elected Foreign Members of the Society.
A paper was read, entitled, " On the Cun'ature of Surfaces." By John R. Young, Esq. Communicated bv John W. Lubbock, Esq., M.A., V.P. and Treas. R.S.
The principal object of this paper is, to remove the obscurity in which that part of the theory of the curvature of surfaces which re- lates 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 cur- vature at an umbihc are not, as at other points on a surface, two in number, or, as had been stated by Dupin, limited; but that they pro- ceed in eveiy possible direction from the umbilic.
The obscurity complained of is attributed to the inaccurate con- ceptions entertained by ]Monge and Dupin, of the import of the sym- bol ^ in the analytical discussion of this question, the equation which determines the directions of the lines of curvature taking the form
at an umbilic. After stating that Dupin has been guided by the de- termination of the diflerential calculus, the author remarks, that in
no case is the diff"erential calculus competent to decide whether 5, the
form which a general analytical result takes in certain particular hy- potheses, as to the arbitrar}' 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 iinal re-
