326 DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 
To express tanr in terms of the amplitude (p. 
Assume the relation established in (13.) or (25.) or (38.) or (39.), tan(p=cos 2 tan/.. 
Introducing' this condition into (43.), we obtain 
etans siiKp cos(p 
tanr= 
or as 
■v/ 1 — sin^)) sin^ip ’ 
^ m — e, \/ w=tan£, 
(44.) 
z=sinjj, 
the last equation becomes 
tanr: 
a/ 
mn sirup cos(p 
sin^ip 
(45.) 
Hence (42.) may now be written 
cl(p 
C Jf 
ij si 
sin^<p 
■tan 
p \/ inn sin<^ cos®"] 
L 
- sin®(p 
_[1 +n sin^ip] V I— sin^ipj V mn, 
Now this formula and (17-) represent the same arc of the spherieal ellipse ; they may 
therefore be equated together. Accordingly 
J (46. 
dip 
[1 +n sin^p] Vl — sin^ip 
? r dip 
sin® 
dp 
1 
■m sin^p] -v/ 1 ■ 
- sin®p 
1 , r sinp cosp"! 
— tan"' I — ^ ' 
p V' mn 
L v'l- 
sin®p 
-J 
J 
(47.) 
This is the well-known theorem established by Legendre, Traits des Fonctions 
Elliptiques, tom. i. p. 68, for the comparison of elliptic integrals of the circular form, 
with positive and negative parameters respectively. These circular forms arise from 
treating the element of the spherical conic either as the hypothenuse of an infinitesi- 
mal right-angled triangle, or as an element of a circular arc, having the same eurva- 
ture. When we adopt the former principle, we obtain for the arc an elliptic integral 
of the third order, circular form and negative parameter. When we choose the latter, 
we get a circular form of the same order, with a positive parameter. Equating these 
expressions for the same arc of the curve, the resulting relation is Legendre’s theo- 
rem. We thus see how an elliptic integral with a positive parameter may be made to 
depend on another with a negative parameter less than 1 and greater than r. 
XVI. We must not confound the angle X in the pre- 
ceding article with the angle X in Art. (X.). Marking the 
latter X by a trait thus,>i;, to distinguish it from the former, 
we shall investigate the relation between them. Through 
ZO the axis of the eylinder, let a plane be drawn making 
the angle with the plane TJdKa. Let this plane cut 
the spherical ellipse in the point a, and the plane ellipse 
the orthogonal projection of the latter in the point Q. 
Through k draw an arc of a great circle /st touching the 
curve, and through Q draw a right line touching the plane ellipse. From Z 
Fig. 4. 
