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Monpay, January 10TH, 1853. 
JOHN ANSTER, LL. D., Vice-Presipent, in the Chair. 
GILBERT Sanv_Ers, 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 :— 
Tf a closed curve B be traced on any surface whatsoever, §, 
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 between 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. 2M 
