DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 385 
H 0 
To determine the value of the fraction which appears under the form of ^ 
when 0=|, we must take the first differentials of the numerator and denominator of 
this fraction. Now, as in (333.) 
Therefore 
■Jo J 
d cot0 
de 
dd 
Vl—i^ sin'^^ 
r 
d^ 
sin^0' 
Hence, when 0=^, 3^=^^ 

dd 
Accordingly 
0 >y 1 - sin^0=^v/ 1 (1 -f2)+ when 0=^. 
JI 
cot 
(a.) 
(b.) 
(c.) 
(336.) 
(337.) 
Substituting this value in (332.), we get 2=at ^dL<P \/ 1 — sin^ip, . . . 
Jo 
the common expression for a quadrant of a plane ellipse, whose major axis is a, and 
TT 
eccentricity i. As it should be, for when 0=^, or n—P, the section of the cylinder is 
a plane ellipse, as shown in Case VII. p. 316. In the spherical form, the limits of 0 
are 0 and while in the paraboloidal form, the limits of 0 are tan"'^^^ and|- 
Section VIII. — On Conjugate Arcs of Hyperconic Sections. 
LVIII. Conjugate arcs of hyperconic sections may be defined, as arcs whose ampli- 
tudes <p, %, CO are connected by the equation 
cosa;=cos^ cos;)(^— sin® sin%\/ l—i^ sin^<y (338.) 
This is a fundamental theorem in the theory of elliptic integrals. 
The angles <p, oj may be called conjugate amplitudes. 
When the hyperconic section is a circle, i — O, and cosft»=cos^ cos%— sin^ sin%, 
whence <u=ip-l-%, or the conjugate amplitudes are <p+x>^ <p and X' development 
of this expression is the foundation of circular trigonometry. 
On the Trigonometry of the Parabola. 
When the hyperconic section is a parabola, i=l, and (338.) may be reduced to 
tan4>=tan^ sec>;^+tan}(^sec(p (339.) 
If we make the imaginary transformations, 
tan<i;= V — 1 sin^y', tan^= V— i simp', tanx^ — 1 sin^', secp=cos®', sec%=cos%'.(340.) 
3 D 2 
