242 Mr. Ivory on the Properties of a Line of shortest 
tently called it in the communication alluded to. I have already 
noticed the inadvertency in this Journal for April 1825; and 
have shown that the accuracy of the solution is not affected by 
it, because the import of the symbol is independent of the name 
given to it, being fixed by the assumed form of the coordi- 
nates. The relation of the arc ¥ to the true latitude may like- 
wise be deduced directly from the equation of the surface, or 
from the expressions of the coordinates, without recurring to 
particular properties of the spheroid, in the manner following. 
From a point in the spheroid, of which the coordinates 
are 2, y, 2, let a perpendicular p be drawn to the surface, and 
extended outward to a point of which the coordinates are @, 
b, c; then, 
p= (a—a2l + (b—yf + (¢— 2): 
and if 2, y, vary in the surface of the spheroid, the condi- 
tion of perpendicularity will be, 
O=(a—a)dx+(b—y)dy + (c —2) dz. 
If the arc w denote the inclination of p to the equator, then, 
c and z being perpendicular to that plane, we shall have 
c—z=p sinwu. Also p cos x will be the projection of p 
upon the same plane; and, as the spheroid is a solid of revo- 
lution, p and its projection will be contained in the meridian 
which makes an angle ¢ with the given meridian to which y 
is perpendicular and z parallel: hence a — x = p cos wu cos 9, 
b—y=pcosusin¢g. The foregoing equation will therefore 
become by substitution, 
0 = cosu (dx cos¢ + dy sind) + dzsinu. 
Now substitute the differentials of the coordinates, then the 
arc ¢ will disappear; and, having divided by cos u cos pd, 
we shall get 
tan wu z 
tan = Vite 
This very simple equation expresses the relation between the 
arcs y and w, of which the latter is the true latitude of the 
point on the surface of the spheroid. It is usual to call the 
arc the reduced latitude; but, as this name is purely arbi- 
trary, it seems preferable to define the same arc from some 
of its geometrical properties. ‘This may be done by saying 
that wu and w are the latitudes of the same parallel to the equa- 
tor, the one on the surface of the spheroid, the other on the 
inscribed sphere. ‘To the formula already given we may add 
the two following resulting from it, which are of continual 
use, viz. < sin u 
sim wy i 
A/ 1 +e? cost u 
cos u 1+eé 
cos) = Swe 
A 1 +? cos? % ‘ Having 
