402 DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 
nj* fit 
or Therefore the arcs have a common extremity. We have also 
tan*X=|^. This property of the plane ellipse, called Fagnano’s theorem, may be found 
in any elementary treatise on elliptic functions. See Hymer’s Integral Calculus, 
p. 209. 
On the Maximum Protangent Arc in the Spherical Hyperconic Section. 
LXXIII. If we assume the expression found for this arc r in (45.), 
mn simp cos(p 
tanr=- 
■ sin^f 
(386.) 
dr 
and differentiate it, as in the last article, and make ^=0, we shall find, as before. 
tamp: 
=— =\/2 
s/ j V si 
sina 
sin/3’ 
(387.) 
If we substitute this value of tan p in the preceding expression, we shall obtain 
tanr= tana secjS— tanjS seca, ....... (388.) 
writing r to denote the maximum protangent. 
Now if we turn to Art. LVIIL, we shall there find that this equation connects the 
amplitudes of three conjugate arcs of a plane parabola. Or if r, /3, and a are made 
the three normal angles of a plane parabola, and (/f.r), (k.(3), (k.ot) the three corre- 
sponding arcs of the parabola, we shall have 
{k.o!) — {k.^) — {k.T)=k tana tan|3 tanr. 
If in (386.) we substitute for sin® and cos<p their values — : and the ex- 
^ ^ '/l+> 
pression will become 
tanr=: 
■v/ mn 
(W)' 
(389.) 
We shall see the importance of this value of r in the next section, 
the spherical parabola, as m=n=i, tanV= 
Precisely in the same manner as in the plane ellipse, we may show that when tanr 
has the preceding value, the arcs drawn from the vertices of the curve have a common 
extremity. This will be shown by proving that the vector arcs, drawn from tfie 
centre of the curve to the extremities of the compared arcs, have the same inclination 
to the principal arc 2a. Now and 4'' being these inclinations, as in XIV., we find 
and (39.) shows that tan<p= cose tan?i. Hence reducing, 
tan^/3 sin®/3 
f 
