406 DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 
expression for the tangent of the inclination of any curve to the plane of xy, namely, 
(i« 
and make we shall get, as before, putting for tan^X=tan^^, the value 
a—b 
Hence the arcs have a common extremity, since they have the 
+ dy 
a d^- 
b' Vdx^ + dy’^~ k 
same inclination to the plane of xy. As |=tan®X is the value of tan% which gives 
the maximum protangent =a — h in the plane ellipse, the base of the cylinder; it 
follows that the point of maximum division on the logarithmic ellipse is orthogonally 
projected into the point of maximum division on the plane ellipse; and the corre- 
sponding protangent in the latter a—b is the ordinate of the parabolic arc, which 
expresses the difference between the corresponding arcs of the former. Thus, while 
the arcs which together constitute the quadrant on the plane ellipse, differ by the 
difference of the semiaxes a—h, the corresponding arcs of the logarithmic ellipse 
will differ by an arc of a parabola whose ordinate is a—h. 
LXXVI. When the amplitude is given by the equation tan^=^^, or when the 
protangent is a maximum, the corresponding arc of the spherical ellipse, or of the 
logarithmic ellipse, may be expressed by functions of the first and second orders 
only. This may be shown as follows. When tan(p=^^ the arcs a and of the 
spherical ellipse, or the arcs 2 and S of the logarithmic ellipse, together make up the 
quadrant C. Hence ff-f (r^=C, or 2H-S=C. But we have also (r=r, as in (52.), 
and S — 2 =t, as in (160.). Therefore 
C-^ 
C + : 
S: 
C+T 
2 =- 
Or (7 and or 2 and S may be expressed as simple functions of C and r. Now C, 
the quadrant, as Ave have shown in the last section, may be expressed by functions 
of the first and second orders only, while r is an arc either of a circle or of a parabola. 
Hence an elliptic integral of the third order, whose amplitude <p=tan”^ 
be expressed by functions of the first and second orders only. 
may 
Section X . — On Derivative Hyperconic Sections. 
LXXVII. We shall now proceed to show that, when a hyperconic section is given, 
whether it be spherical or paraboloidal, we may from it derive a series of curves, whose 
moduli and parameters shall decrease or increase according to a certain law ; so that 
ultimately the rectification of these curves may be reduced to the calculation of 
circular or parabolic arcs, or in other words, to circular functions or logarithms. 
We shall also show that all these derived curves, together with the original curve, 
may be traced on the same generating surface, i. e. on the same sphere or para- 
boloid. 
