341 



radii of a sphere be drawn parallel to the radii of absolute 

 curvature of any closed curve whatsoever, they will divide the 

 sphere into two equal parts ; for the proposed curve may be 

 regarded as a geodetic line upon a surface so described that 

 the tangent plane at any point along the given curve is per- 

 pendicular to the radius of absolute curvature at that point. 



If the boundary curve be a loop of a geodetic line, the 

 total curvature of the included portion of the surface is equal 

 to a hemisphere diminished by the external angle of the loop. 

 If the boundary be a polygon whose sides are geodetic 

 lines, the total curvature will be equal to a hemisphere dimi- 

 nished by the sum of the external angles of the figure. This 

 proposition includes Gauss' celebrated theorem respecting the 

 total curvatm*e of a triangle formed on any surface with geo- 

 detic lines. 



If the surface S be itself a sphere, we can represent the 

 area of any closed curve B traced upon it by a plane angle. 

 For this purpose, let a developable surface be circumscribed 

 along the curve B, and let the angle be constructed as in the 

 theorem. In this way we find the area of a small circle of the 

 sphere to be equal to the defect by which the developed angle 

 of the circumscribed cone falls short of four right angles. 



The Rev. Professor Haughton communicated the follow- 

 ing account of some barometric determinations of height made 

 by him, with the view of examining by direct observation the 

 different formula? which have been proposed for introducing 

 the hygrometric condition of the air into the calculation of 

 heights : — 



The uncorrected barometric formula is the following: — 



H«10000/«»-(l + JL)logiL (I.) 



in which 9 denotes the mean excess of the temperature of the 

 2m 2 



