DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 399 
Now as <p, Xi and a are conjugate amplitudes, 
See Hymer’s Integral Calculus, p. 206. 
Whence |[2«— 2;^— 2^] =-2!^[n<I>+wX— wH— sinip sin% siniy] 
(»■) 
We have now to compute the sum of d>+X— H. 
/z^r- cosis cosy — cosw sinco cosw'V^Isy _ sin^co (cos^o) — V) .. , 
Since \/I»=~ — j =-5 — =£2=— tvt-ft , ii we make, as 
^ * sin® siny 1— nsin^co iN„U ’ ’ 
psin^p cos^p 1 
, sm2;)^cos2x 
sin^o) COSTCO”] 
1 ^1 
"nsin^w ^ 
, ^sin^x , 
1 rasin^ippi 
1 — ; 
j 
^ N 
X 
" N, J 
1 ul 
- ' 
1 N ^ 
^ X 
J- 
before, cosip cos}' cos<!y=V, and sin^ sin%; sin4>=U. Finding similar expressions for 
O and X, we shall have 
Now 
and 
whence 
n sin^if) cos®<p cos^<p(l + n sin^ip — 1) cos^ip cos^ip 
UN = NU “ NTJ 
cos^^ 1 +n— nsin^(p — 1 
(1-n) 
NU 
n sin^<p cos-'ip 
nNU 
cos^^ (1~^) 
■«U~ nNU ’ 
NU 
'nU 
u 
j Vn sin^tp V V 
^iNTT’ NU““U~NU' 
Finding similar expressions for the functions of a and %, and recollecting that, as 
in (362.), cos^(p+cos^>^+cos^(y=l+2V— we shall have, making W=l — w+wV, 
wU(nO+wX — wr2)=3— w+nV — W 
Now whence 
«U l^nO+wX — nVL — ^Jcl<P\/l +Jdx,>/ 1 — J ^ 0 * 
We shall find, after some complicated calculations, N^N;j;N„=W^— . (370.) 
and N;^N„+N.N^+N^N;,=W^+2W-?^(l-^^)(^^+w^)U^ (371.) 
Substituting the values hence derived, the whole expression becomes divisible by 
nU% and we shall obtain, finally, the following expression, 
A/mnr I -fT ^ .2-1 TT n 'v/xWU , 2mn^ -v^xUV /'q'70 ^ 
[nd> + wX — Wf2— ^"U]=:^™ 2-TT2 + 7 WW2 (372.) 
n—m W^— n-^xU^ (n— m)(W^ — n-'xU^) 
It may easily be shown, that 
n + nV + n xU"| 
n + nY—n V xUJ ’ 
(373.) 
3 F 
MDCCCLII. 
