52 PHILOSOPHICAL SOCIETY OF WASHINGTON. 
size. The results here deduced may therefore be considered to be 
results of a comparison of the circumcircles of these two triangles. 
A corresponding study of the relations of the tangent circles (in- 
and escribed) would therefore be expected to yield many more 
properties as there are four times as many circles to be considered. 
Concerning the phrase “bisection in the extended sense,” Mr, 
Dooxirr Lez suggested that the term “sesquisection ” might be advan- 
tageously employed. As to the name “nine-points circle,” Mr. 
KuMmMELL said that it was plainly defective, either “ six-points” 
or “twelve-points ” circle being satisfactory, according to the point 
of view taken, but that nine-points circle was not a correct desig- 
nation from any point of view. Further remarks were made by 
Mr. Exxiorv. 
Mr. C. H. KumMe tt presented a communication entitled 
DISTANCES ON ANY SPHEROID. 
[Abstract. ] 
The present form of solution of the problem to determine the 
shortest distance between two points on a spheroid which are given 
by their latitudes and longitudes is characteristic in making use of 
the Gaussian algorithm of the arithmetico-geometric mean. This 
and a corresponding transformation of the amplitudes give the 
necessary elements for computing in three terms the distance precise 
to the eighth order at lJeast. The form is also remarkable for its 
symmetry and easy extensibility to still higher precision. 
Also the preliminary part of the problem in which the excess of 
the spherical longitude over the spheroidal is determined by succes- 
sive approximations is much facilitated by the introduction of an 
angle 7, which is closely related to ¢, the angle of eccentricity, and 
which varies between 0 and «. It was thus possible to express the 
excess of spherical over spheroidal longitude in one term, precise to 
the 6th order at least. 
[This paper has been published ig full in the Astronomische 
Nachrichten, No. 2671.] ; 
In reply to a question, Mr. KuMMELL said that the®rdinary for- 
mulz for computing distances between intervisible points on the 
terrestrial spheroid are all that can be desired. The formulse here 
