212 Boyal Society. 



animal and a powerful Leyden batterjs it was concluded that the 

 quantitj?- 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 every 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 actioii and re-action of the electi-ic 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*. 



Dec. 20, 1838. — A paper was read, entitled, " On the Curvature 

 of Surfaces." By John R. Young, Esq. Communicated by 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 umbilic 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 every 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 differential calculus, the author remarks, that in 



no case is the differential calculus competent to decide whether -, the 



form which a general analytical result takes in certain particular hy- 

 potheses, 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 re- 

 sult, to whicli they lead, equally indeterminate ; and that, therefore, 



when such result assumes the form -, 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. , 

 In a " Memoire sur la Courbure des Surfaces," (Journal de I'Ecole 



* 111 the First Series of the Philosophical Magazine, vol. xv. p. 12G, 

 will be found a translation of E. Geoffrey's Memoir on the Anatomy of 

 the Electrical Fishes, including that of the Gymnotus. — Edit. 



