124: PHILOSOPHICAL SOCIETY OF WASHINGTON. 
sidered straight all through at any of its points. This curve may 
be considered the ideal curve of a primary base line. Various 
names have been given to it when on the terrestrial spheroid. Dr. 
Bremiker, who appears to have first considered it (in his Studien 
ueber hoehere Geodesie, 1869), proposed the name “ Feldlinie”; 
that is, field line. He thinks it should be adopted as the geodetic 
line, because both linear and angular measurements conform to it. 
Clarke, Zachariz, and Helmert have also mentioned it, the latter, 
however, only in a note, where he remarks that it deserves no con- 
sideration in geodesy. 
To the second class belong two curves: 1. A curve described as 
follows: The surveyor at the starting point takes his directions 
from a staff at short distance and directs his assistant to place 
a staff in the prolongation. Repeating this operation from the 
first staff, from the second staff, etc., he describes a curve which 
is well known to be the shortest curve between any of its points. 
It is usually called the geodetic line. However, since this name 
would apply at least equally well to the three curves already con- 
sidered, I propose the name brachisthode (fpaytoros). The proper- 
ties of this curve need not be considered here, such mathematicians 
as Gauss, Hansen, Bessel, and others, having perfected its theory. 
Helmert, in his “ Hoehere Geodesie,” makes this curve the basis of 
nearly all geodetic computations. The brachisthodic process on a 
plane evidently results in a straight line, and on a sphere in a great 
circle. If, on these surfaces, it is in starting directed to a distant 
point, that point will be reached (disregarding errors of observation). 
Not so on other curved surfaces; there, in general, the first element 
of the brachisthode is not in direction to any of its points at a finite 
distance. 2. The loxodrome being a curve which has a constant 
inclination to a given direction, may, perhaps, be mentioned as be- 
longing to this class. 
The general equations of the two-end curves on any surface may 
be developed as follows: 
Let the equation of the surface be: 
u=f (2%, y, 2) = 0 (1) 
then if (€, 7, £) is any point in the normal at the surface point 
(x, y, 2), we have its equations: 
ae a @) 
(i) (%) (z) 
