DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 347 
If we put c for the eccentricity of the plane elliptic base of the cylinder, we shall 
have after some obvious reductions, writing- for the complement of c, 
(1— /")(l-c") = (l— or fj=\—n (131.) 
Now this simple equation between n, i and c enables us with great ease to deter- 
mine the eccentricity c of the base of the elliptic cylinder, whose section with the 
paraboloid gives the logarithmic ellipse, when we know the parameter n, and the 
modulus ^, of the given elliptic integral. 
If we reduce this equation, it becomes the denominator of (130.). 
XXXVI. To integrate the expression given in (127.), we must assume 
^ _sin<pcos<p A/l-i2sin2(p 
" [l-?ism2<p] 
Differentiate this expression with respect to <p, and we shall have 
1 — 2(1 + sin^(p + 3z^sin^(p ^ 2ra (sin^(p — sin‘^<p)(l — sin^<p) 
df 
(a.) 
[1 — nsin^<p] Vl— [1 — n sin^<p]^ Vl — sin^(p 
Let 1— wsinY=N, 1 — i^sin^ip^I, as before. 
Separating the numerators of the preceding expression into their component parts 
and attaching to each their respectiv^e denominators, we shall have 
2(1 +i^) sin^(p 2(1 +2^) (1— nsin^f— 1) 2(l+i^) 2(1 + *^ 
N VI 
The next term gives 
3?^ (1 — 7isin^(p — 1) sin^ip 
N VI 
32^ sln^<p 3i^ sin^^ 
n VI wN VI 
7r-\ 
N VI n N VI w VI ' wN VI 
Now these two terms may be still further resolved ; for 
3i^ sin^ip 3 (1 — — 1) 3 VT 3 
(C.) 
(d.) 
n VI ^ VI 
32^sin^^ 32^ (1 — rasin^ip — 1) 
, and 
n n V'V 
3i^ . 
nNVl 
whence (d.) becomes 
NVi 
32^ sin-^ip 3 Vl 
NVI —~;r' 
222VI'^722NVI’ 
322 . 3^ 
(e.) 
72 Vi 72^VI ‘ 72^NVI 
Combining the expressions in (b.), (c.), (d.) or (e.), the first term of the second 
member of (a.) may be written 
[l-2(l+2'^) sin^^ + 32^sin^^] 3VI ■ r2 . ^ §1 J__L fl —-/'l 
[1 — 72 sin^ip] Vl — [_72' / n 72 JVI ' L 72 ^ JN VT ' 
The second term, of (a.) may be thus developed, 
2I . 2I 
(1—72 sin^ip)^ 
272 sin®<$>Vl 2^(1 — 72sin^ip — l)Vl_ 
72 TV 
and these two latter expressions may be written 
2 22’^ (1 — 72sin®<p — 1) 
2l 2(1 — 2^sin®f) 
“nvi~ nvi ' 
’nVi'^NVI ’ 
22^ 1 . 22^ 1 
(g-) 
NVI 
NVI 
72 Vl"^ n NVI NVI ’ 
