348 DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 
, / \ L 2ra sin^a 
whence (g.) becomes — = 
m. 2nsin%I , 
The term — may be written 
2i^ 
'Vl 
/ ^^\ 1 
21 
2nl sin^<p 2l pi — 2?i sin^tp + sin^^— 2 + 2n sin^<p + 1 j 2l , 4l 
N2^I 
Now 
and 
whence 
n L 
21 
41 
NWi 
nl 2 
4(1 — i^sin^ip) 4 4i^ (1 — n sin^(p — 1) 
n.X^v/l 
(h.) 
• (k.) 
nN'V^I 
41 
/iNVl' 
/iN^I 
4i^ (i^—n) 
4 ^7q 
n^i^I 
nNVl 
1 
Nv'f 
N'/I 
Combining (k.) with (m.), we shall have 
2nl sin^tp 
2 V'l 
4i^ 
nwt 
n 
► 
2^1 , 
'4i^ 
2?\ 
1 , P 
n ' V 
n ) 
y'l+i 
21 
1 1 
lV 
Jn^/1 
) I 
\ n j 
'nWi ’ • 
\ 
i! 
1 
d?) 
n 
N^VT 
(ra.) 
(n.) 
(p.) 
(q-) 
(I--) 
adding (n.) to (h.), 
2w(sin^(p — sin'^ip)! 
N^Tf = 
adding (f.) and (p.) together, we get as the final result, 
d<P n n J -^Nv'l 
or multiplying by n, transposing and integrating, 
But we have shown in (129.) that 
n{^ — ra)(l — n) ^ ^ 
, 2\2n — ? — ^ rr . — 7i\ Cd<P r2n— 2 ^— «' 
whence 
Vn.{f 
Hence, an arc of a logarithmic ellipse may be expressed by a line O,,, and in terms 
of elliptic integrals of the first, second and third orders ; the latter being of the 
logarithmic form (127.) may be written in the form 
fd(p[l-i^sm^(p] 
k \/C(A + B)J[l-wsm2<p]Wl-i2sin2<p ' 
XXXVn. When the cylinder and the paraboloid are given, we may determine the 
parameter, modulus and constants of the functions which represent the curve of inter- 
section of these surfaces, in the terms of the constants a, h and k. 
The modulus parameter, coefficients and criterion of sphericity may be expressed, as 
linear products of constants, having simple relations with those of the given surfaces. 
Resuming the equations given in (125.), 
AC=b^k% C-A=k^-\-a^-b% B=a--b% 
df 
(133.) 
