408 DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 
We may transform (395.) into 
J [1 +n^ sin^\I/] 
Now 
Hence 
mi. r . _ , mi, m + n 
and 2— 7W+— '= . 
n' • -w n 
n, n" 
(397.) 
or dp 
{m + n)V ji/f 
1 rd^/ 
JmXi 
mn,V mn^ 
|[1 -f- sin^4/] Vl/ 
2 
mn\ V I 
tan ‘(^y^?Jsin■<P). (398.) 
We shall now show that v Anw sin(p_ ^ (399.) 
vl—i'‘siny ^ ‘ 
If we revert to (189.) and (193.), we there find 
2 sin^ cos<p=sin%^[-\/l^+?!^ cos-v^/], and 2^/l = (l+J)[x/l,+^ COS-v//] . 
(396.) gives >/ mn=^ni{ 1 +j) ; therefore sin-^/. 
V 1 
If we replace in the preceding equation by its valuej-^, and put for 
1 sin^-v^/, 
^ - - 7^ tan- [- 
V /jMvI mre V TOnjN^vb ^mn L 
^ mn sin(p cos^q 
VI 
J- • 
(400.) 
Now the common formula for comparing circular integrals with conjugate para- 
meters is, we know, see (47-), 
/I 
fl—m^ 
Cdp iU 
_ 1 _ ^ (nn-l 
P Vmn sinpcospq 
V « / 
JN VI ' 
\ m ) 
JmXi mn\ 
1 VI^ Vmn 
L Vl—i'^ sin®p J 
Adding these equations we obtain this new formula 
By the help of this important formula we may establish a simple relation between 
the sum of the original conjugate functions of the third order, and the first derived 
function of this order. 
LXXVIII. If (T be the arc of a spherical ellipse, it is shown in (46.) that 
. 1 . 
and in ( 17 .) that 
/l+ra\ . — C dip f*d<p _ 1 r sinp cosa' 
’'-y n L'Vr=:Fih^"'J 
M VI 
Adding these equations together, and introducing the relation just now established, 
[m + n] ^,—^ dvf; 
mn 
Nj VI/ V mnj VI 
'd<p 
— tan' 
Vmn sinp cosp' 
t v mn s 
VT^ 
■i^- sin-<p J 
Now as m — n = i^ — mn, {ni-\-nY=‘V —2i^77m-\-m-rf -\-Amiu 
( 402 .) 
