400 DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 
or writing, as before, W for 1 — w+wV, and multiplying numerator and denominator 
by the numerator, 
d<p 
N,^I 
Now let 
r d Y r dw 1 , r W + wa/xU t ^ 
(3/5.) 
^/xU 
w 
= sms 
and the preceding logarithm becomes log(sec|+ tan|), which is, we know, the integral 
jcosf" 
n \/xWU _ 2|’^~=sec| tan|+J^ 
of 
Now 
we shall have, dividing by 2, 
2.-S,-2,=4Jj4 
'AL 
Icos^’ 
kmn^ xUV 
(376.) 
Hence the sum of three arcs of a logarithmic ellipse may be expressed by an arc of 
parabola and a right line. 
When one of the arcs is a quadrant, V=0, and the equation becomes . 
which coincides with (160.). 
[2f-2.]-2,=4Jf (3-.) 
If we apply to (163.) the same process, step by step, and make sin^=^^C^'— , in 
which W,= l— w+mV, we shall have 
VV, 
krn^n -y/x^UV 
- m) ( — OT^x^U®) 
If we subtract this equation from (376.), we shall have 
. (378.) 
Jcos-^T ' JCOS'^Ty JCOS^Tii '' ' ' 
Jcos^^^j 
fdr , 
4-1 
1 dr, 1 
fdT,, mn 
r m V n Vx 
|cos®^ 
'COS^T '^J 
ICOS'^T, ' 
|cos^T^^ ' n — m^ ' 1 
[y^y-mx^]^ W^-zi^xU^J 
Now this last term is divisible by (n—m), and may be reduced to the expression 
mn i/ mn\]\ \y ^ — V)^] 
(380.) 
[W^-n^xU^] [W,"-m%U2] 
If in (170.), which gives the relation between conjugate elliptic integrals of the 
third order, we substitute successively <p, % and u, and add the equations thence 
resulting, we shall have 
rii+r.3x 
Jcosj ' Jcos^ JCOST ‘ JCOST^ JCOST,/ ' ' 
in which 
sm? 
V " mn simp sinj^ sinw 
sin^ 
V ” mn sinp sin;)(^ sinco 
1 + 
1 —n 
COS<p COSp^ COSC 4 ) 
1 + . 
, */ mn sin<s cosp 
sinr— , sinr = 
vl— t^sinp ‘ 
1—m 
i'^mn sin;^ cos;;^ 
% 
cosp COS^ COSCO 
. V' mn sinco cosco 
smr^,=- 
( 382 .) 
