Distance traced on the Surface of an oblate Spheroid. 245 
further the line is produced. If therefore we suppose that 
the variable parallel to the equator begins to move from the 
two initial points, passes beyond the equator to the extreme 
latitude — z, then returns to the other extreme latitude + 2, 
and lastly, falls down to the situation it first left; the moveable 
point in the oblique great circle will have made a complete 
circle in longitude, and will have returned to its original place; 
but the moveable point in the geodetical line, although it will 
have returned to the same latitude it left, will not have com- 
pleted a circle of longitude, and therefore it will meet the pa- 
rallel of latitude in a point different from its first place. 'Thus 
a geodetical line dves not return into itself; if it be continued 
for several successive circuits of the spheroid, it will form a 
spiral line upon its surface. It is manifest from the analytical 
expression of d ¢, that when z, which is the limit of {, is very 
small, the whole arc of longitude answering to one turn of the 
: a 60° ° 
geodetical line, approaches very nearly to See. or, if we 
estimate the longitudes on the equator of the spheroid, it will 
be equal to an arc of the same length with the periphery of 
the inscribed sphere. Accurately speaking, the arc mentioned 
is the limit of the longitude made in one turn, when the geo- 
detical line cuts the equator in an indefinitely small angle. It 
follows therefore that the equator itself is not comprehended 
in the analytical expression of the arcs of shortest distance ; 
but, when the inclination to the equator is infinitely small, all 
the turns of the spiral curve become blended with one another 
and with the equator. 
We may next compare the lengths of the geodetical line 
with the ares of the oblique great circle cut off by the same 
parallel to the equator. ‘The first formula (A) shows that ds 
is greater than ds'; and hence an entire turn of the geodetical 
line is greater than the periphery of the great circle. But 
these two quantities approach nearer to an equality as the 
obliquity to the equator increases; so that they are exactly 
equal when the geodetical line makes an infinitely small angle 
with the equator. This agrees with what has already been 
said respecting the longitude in the same circumstances. 
From the same expression it follows that all the turns of 
the same geodetical line are equal and similar; and even that 
every single turn consists of four equal parts or quadrants: 
for the integral of ds has the same value, while varies be- 
tween the limits 0 and + 7. 
Let a denote the are of the oblique great circle between the 
initial point and the equator; then a —s! will be the are be- 
tween 
