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 curvature 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 formule which have been proposed for introducing 
the hygrometric condition of the air into the calculation of 
heights:— 
_ The uncorrected barometric formula is the following:— 
98 p 
fath. 
H = 10000 (1 + 133) log + (1.) 
in which © denotes the mean excess of the temperature of the 
2m 2 
