352 DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 
(148.) may now be written 
Z-2=jd?ix/P +Q' sin'?.+R' (I57.) 
If we measure the arc of the logarithmic ellipse from the minor principal axe, or 
from the parabolic arc which is projected into h, instead of placing the origin at the 
vertex of the major axe as in (119.), we must put 
x=0'sin3^, «/=&cos^; (158.) 
and following the steps indicated in that article, we shall obtain 
^S=Jd&\/P'-|-Q' sin^^+R' sin^S- (159.) 
If we now make ^=X, and subtract the two latter equations, one from the other, 
the resulting equation will become 
(*60.) 
But this integral is, we know, the expression for an arc of a common parabola, 
whose semi-parameter is k, measured from the vertex of the curve, to a point on it, 
where its tangent makes the angle r with the ordinate. 
Thus the difference between two elliptic arcs measured from the vertices of the 
curve, which in the plane ellipse may, as we know, be expressed by a right line ; and 
in the spherical ellipse by an arc of a circle, as shown in Art. XV. ; will in the 
logarithmic ellipse be expressed by an arc of a parabola. As a parabolic arc can be 
rectified only by a logarithm, we may hence see the propriety of the term logarithmic. 
by which this function is designated. 
XXXIX. If from the vertex A of a paraboloid, an arc of a parabola be drawn, at 
right angles to a parabolic section of the paraboloid, it will meet this parabolic sec- 
tion at its vertex. Let the arc AQ be drawn at right angles 
to the parabolic section Qv of the paraboloid, the point Q 
is the vertex of the parabola Qv. 
Draw QT and tangents to the arcs QA and Q^;. Then 
QT and are at right angles. As QT is a tangent to a 
principal section passing through the axis of the paraboloid, 
it will meet this axis in a point T ; and as is a tangent to 
the surface of the paraboloid, it will be perpendicular to the 
normal to the surface QN. Now as is perpendicular to 
QT and to QN, it is perpendicular to the plane QTN which 
passes through them, and therefore to every line in this 
plane, and therefore to the axis AN, or to any line parallel to it, as the diameter 
Qn. Hence, as the tangent Q^ to the parabola Qy is perpendicular to the diameter 
Qn, Q is the vertex of the parabola. 
Hence in the logarithmic ellipse, one extremity of the protangent arc is always the 
vertex of the parabola which touches the logarithmic ellipse at its other extremity. 
Fig. 15. 
