DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 373 
LII. By giving a double rectification of the common hyperbola, we shall the 
more readily discover the striking analogy which exists between this curve and the 
logarithmic hyperbola. 
Let U be an arc of a common hyperbola, whose equation is 
Assume 
cos^A 
sin^ A 
^ fl^cos^A— i^sin^A’ y a® cos^A — sin^A 
Differentiating these expressions, and substituting, we get 
dU }?■ . . „ G- + Z»® . 1 1 -o 
r , nr Assume smV= -jj- sin% and let • 
’ -sm'^A ' 
a\ 1- 
(a.) 
(b.) 
Finding from this equation the value of as ^=^’^5 shall finally obtain. 
*2 
since 
fl(l — 
« * J [1 — sir 
d<p 
(C.) 
sin^ip] -/l— i^siiKp^" 
(31.) gives — U=Jj9d>.+J^dX, or U=— J;?dA— ^ (d.) 
XT 2 2 2^ 72 • 2-, + sinAcosA . G® • 2 -. / \ 
Nowas/=a-cos’X-i''sin=?,, and as smV=^.sin% (e.) 
dp + y 
dA= 
G COSp 
and as p= a cos<p, 
(f.) hence ^=—\/a^+&Han^\/l— i^sin^ 9 ; . . (g.) 
pdX= 
dA' 
G® COS^ipdip 
{1 +z^ — z^sin^ip — Ij- 
(g^ + 6^) v/ 1 — sin^ip i s/X—P sinf 
whence, finally, tanip^/l— jcl^\/ld-(l 
This is the expression for an arc of an hyperbola referred to in (XLIX.). 
m . 1 r da 1 Tl /7 
The integral 
See Hymer’s Integral Calculus, p. 195 . Adding this integral to {k), 
iu+(. -if^j = (l_i.)J-^j+ tan^yi-!!^” . 
. . (h.) 
(k.) 
VI ' ' 
but 
V I v\ 
IT !• . 1 . 1 / .o^ *U , r da tana , Fd<P 
Hence dividing by (1-1*), i;(r:^+Jr 7 l= Vl+J vT’ ' ' 
(m.) 
(n.) 
but (0.) gives 
Eliminating U from these equations, we obtain 
r dip ^ r dip r dp 
J [1 — sin^f ] V I— sin“^<p J [1 — sin^<p] Vl—i^ sin^ip J -v/ 1 — si 
tamp 
= 4 . . 
2 ^ a/ 1 — z^ sin®<p 
( 293 .) 
