DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 325 
We must first show that this amplitude x, is equal to the amplitude (p in (13.), and 
therefore to in (25.), as we proved in (X.). 
In an ellipse, if -ip and X are the angles which a central radius vector, and a per- 
pendicular from the centre, on the tangent drawn through its extremity, make with 
the major axis, we know that tan '4/=-^ tanX=£^^ tan?i. Introducing this value of 
tan-p into (6.) and reducing, 
„ „ „_r tan^a cos®A-f-tan^j3 sin®X i 
COS p = cos a COS |3 7 — 2 2 , , , — ^ 2 — • 
^ " [tan'^a cos^p cos \ -t- tan^'p cos^x sm^Aj 
Comparing this value of cos^p with that assumed for cos^p in (13.), namely, 
2 sin^a cos^(p -1- sin^/3 sin^<p 
COS p cos^cp -l-tan^/3 sin^ip’ 
we get, after some reductions, 
tanip=coss tanX . (39.) 
But in (38.) we assumed tan%=c 0 s 2 tanX. Hence the amplitudes p, p and % in (13.), 
(25.), and (38.) are equal. We may accordingly write p instead of %. Substituting 
the value of tanX, derived from the equation tanip=coss tanX, in (38.) the integral 
in (3/.) becomes 
C cos« cos/3 [sin^« — (sin^« — sin^/3) sin^<p] dp 
J [cos^«+ (sin^a— sm^/3)sin®p] -v/ sin^acos^^p + sin®/3sin^p’ 
Now 
COSJ: 
COS « 
'cos /3 
sin^a — sin^/3 . 
cos^« 
, tan^£= 2 sin^;?= 
sin^a - 
• sin^/3 
sm^cc 
(40.) 
Making the substitutions suggested by these relations and reducing, we get 
COS/3 Cr 
dp -1 
cosa cos/3 ( 
^ dp 
' cos« sinaj [ 
_ [1 + tan^e sin^p] V 1 — sin‘^»] sin^p J 
sina J 
Vl — sin^») sin^p ’ 
an elliptic integral of the third order, with a positive parameter, and therefore of the 
circular' form. 
This is case IX. in the Table, page 316. 
Writing n for tan^s, i for sin;?, and expressing sinci, cos«, sinjS, cos(i in terms of 
n and i, (41.) becomes 
f <'f 1 _ AL f— . 
\ ^ y J L[1 sin^p] -Cl — z^sm^pj -v/mre J ■/ 1 — sii 
XV. To express the protangent r in terms of X and p. We found in XII. 
‘ sin^p 
(42.) 
tanV= 
>2^2 
f^p 
(A^— sin^A cos^A 
p2 p2^2 \jp. ^ ^2 cos^A - 1 - sin^A] \o? cos^A -f sin^A] ' 
Now 
whence 
A=^tana, B = ^ tan/3, 
„ A^— , . „ sin^a — sin^/3 
and sin‘£= 
tanr= 
^ sina sinA cosA 
Vl — sin^A V 1 — sin^s sin^A 
(43.) 
