DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 361 
sin 20=2 sin0 cos0=cos4/, and 2d9=d4^. Therefore (202.) will become 
4d22 
+l(F-i) (2*4.) 
hence as A=|- we shall have 22= v'a^+A^Jd'vjl/^Xl — . (215.) 
This is the common form for the rectification of a plane ellipse, whose principal semi- 
axes are y/a^-\-h^ and a. Let be the eccentricity of this plane ellipse, 
h 
B 
1 - Vl 
‘ -/aHA2~2A-FB 2-n 
and the relation between (p and is given by the equations 
26=^-\-4^, tan^0=;r-^-D tanV, or tan0=\/l — w tan<p. 
(216.) 
Hence 
A+B 
r=S^=(l-”)tanV. 
(217.) 
1 TT 'TT “TT 
When 4'=0, ta.n<p=-^; when •4/=2’ 9=2’ '^= “ 2 ’ 9=^- Hence 4' is measured 
from the perpendicular on the tangent to the ellipse, at the point which divides the 
elliptic quadrant into two segments whose difference is equal to a — b, as will be 
shown further on : while <p is measured from the semitransverse axe a. Thus while 
■v// vanes from — g (that is from the position at right angles to this perpendicular, and 
below it,) to 0, that is to the perpendicular itself, <p varies from 0 to tan“'^; and 
TT X “TT 
while 4> varies from 0 to 9 varies from tan“*^ to -. Thus while -^p passes over 
two right angles, <p passes over one right angle. 
We may now equate the two expressions (211.) and (215.), 
jdspj l-i‘ sin^^=^=^[^d9yi+<V([-^-<J>]^ 
(218.) 
or we may express an elliptic integral of the first order by means of two elliptic in- 
tegrals of the second order. Thus we obtain the geometrical origin of this well-known 
theorem. 
When the functions are complete, since 
ri 
d-v^x/ 1 — sin^-4/=2\d’».^\/ 1 sin^-v//, we get 
5i 
|d-v^x^ 1 — sin^4'=*/ 
which agrees with (186.). 
I— 
1 
d<pJl + { 
l-wxf" d^ 
'J\ VI 
(219.) 
3 A 2 
