412 DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 
\/ » 
Now in (399.) it has been shown that\/ sin\p = — and as mn= 
the last term of the preceding equation may be written 
'•_d r V mn sin(p cosf 
vT 
]d?5 
v/ 
mn ' 
mn sin^ip cos^^ 
Substituting this value in the preceding equation and comparing it with (169.) or (I70.), 
we shall find 
f\ —m\ 
if 1 
fl-n\, 
P dip (ji — m) 
V / 
JM v/l 
1^4 
JN V 1 nm 
2 ijuyi, 
This equation is analogous to (401.). By the help of it and the last equation we can 
always express 
f d<p (* d(p . p r ‘i'l' 
\vr VT ov Kv - - v in terms of \— — 77- 
JM^/I JNVI 
Since symmetrical with respect to n and m, we should have obtained the 
same value for the derived parameter had it been deduced fromJ ^j^^^j instead of 
Since 
mn 
(1 —jY—mn 
"'/-(I +7)2’ 
—mn r ^ \—m— \—n~\^ 
, or n.= , y=\. . . . (41/.) 
-mn’ ‘ L'v/l-m+ V 1-nJ 
LXXXI. We may express and simply, in terms of a and h, the semiaxes of 
the base of the elliptic cylinder, whose curve of section with the paraboloid is the 
logarithmic ellipse. 
In (171-) we have found the values of m and n in terms of h and k, namely. 
« \^mn[\—m) ^ \/mn{\—n) 
k n — m k 
n—m 
(a.) 
t — b V"l — m — v^I- 
a-bV 
Hence oi- assuming the value of n, in (417.) »,= 
Now w— m = (l— m) — (1— w) = (v/l ~m + n/ 1— ^)(\/ 1 — m — 
Or as »,=7T^„ 1 -m,=^ +jY-mn^{yT=m+ 
' (l+jf’ ' (1+7)' (1+7)' 
, , . . a — b ^ mn 
and (a.) gives 
therefore 
k V\ — m + -V^I — n 
1 (^ I — m — V I — nY 
. Hence reducing, ..a 
m, mn ‘ k^+{a — b)- 
If we now compare together these expressions for and n^, namely, 
{a—bY /a — b^^ 
[a— by /a — b\^ 
F + (a-6)2’ ^i—\J^b)’ 
we shall find that ny>m^, so long as k>2\/ab-, that when k=2\/ab, my=-ny, 
and that when k<2\/ ab, 
