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Monday, January 10th, 1853. 



JOHN ANSTER, LL. D., Vice-President, in the Chair. 



Gilbert Sanders, Esq., was elected a Member of the Aca- 

 demy. 



On the recommendation of the Council, it was Resolved, — 

 " That leave be given to read Papers of which the general 

 nature shall have been approved by Council, but that, unless 

 an Abstract of a Paper shall be delivered to the Secretary of 

 the Council, on or before the night of reading, the title only 

 of it shall be published in the Proceedings of the Academy." 



A letter from Mr. Macaulay, returning thanks for his elec- 

 tion as an Honorary Member, was read. 



The Rev. Professor Graves communicated the following 

 theorem relating to the total curvature of bounded portions of 

 surfaces : — 



If a closed curve B be traced on any surface whatsoever, S, 

 the total curvature of the included portion of the surface may 

 be represented by means of the following construction : — Let 

 a developable surface, D, be circumscribed along the bounding 

 curve, and let it be opened by cutting it along one of its recti- 

 linear generatrices, G, and developed upon a plane ; then the 

 angle behoeen g g, the two right lines which correspond to 

 that generatrix, will represent the total curvature of the pro- 

 posed portion of the surface. 



To prove this theorem, let us conceive a sphere whose 

 radius is unity. Let a cone, C, be formed by radii paralel 

 to the rectilinear generatrices of the circumscribed developable 

 vol. v. 2 m 



