398 DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 
tion between the five circular arcs 0, 0^ r, 
0 + 0^=r+r — 
We may give an independent proof of this remarkable theorem. 
The primary theorem (338.) cos&;=cos 9 cos)( — simp sin 
(360.) 
gives 
sinw cosw sin(p sin;!^^ sinw cosw 
VY cosip cos;j^ — cosco 
and cos^<p+cos\-l-cos*ii;= 1 +2 cosp cos)^ cos4>— f sin^p sin^)( sin^<y. 
Let 
Now 
whence 
and 
whence 
tan(r+r,— rj = 
sinpsinxsina;=U, cosp cos^ cosa;= V (361.) 
Y mn sinw cosco ^ mn U cos^w 
tanr„=: 
V\-i 
■ sin'^w 
COSTCO— V 
. mnUcos^a , V^UcosV- 
tanr= — tanr = = , 
COS^p — V ^ rnsV — V ’ 
cos^;)(^ — V 
, . tanr + tanTy—tanT^; + tanr tatiTytariT;, 
’■/;) 1 + tanr^, tanxy + tanr tan^^— tan tanr ’ 
— r cos^p cos^% cos^w mwU^cos^p cos^;^ cos'-w i 
[_cos^p — V~^cos^p^ — V' COSTCO — V (cos^p — V)(cos^% — V)(cos®co— V)J 
-.2 r cos^;)(^ cos^'cu COSTCO cos'^p cos^p cos-% 1 
1_ (cos‘^;)(; — V) (COSTCO — V) ' (cos^co — V) (cos'^p — V) (cos^p — V) (cos-;;^ — V) J 
If we reduce this expression, we shall have, on introducing the relations 
cos^p+cos\+cos^ft;=l+2V— | 
and cos^(y cos^^^+cosV cos^&»+cos^x cos^p=V^+2V4-J^U^,i 
tpnr<rJ-r_^ [2/ + (iH V] V mnY _ 
T / „) 
(362.) 
(363.) 
If we now combine the values of tan0 and tan0p given in (355.) and (358.), we 
shall have 
tan(U+UJ + ^ 
whence 0d-0^=r+r^— 
as is evident from an inspection of the preceding formulae. 
On Conjugate Arcs of a Logarithmic Ellipse. 
LXXI. In (162.) substitute % and a successively for p. Let 
\ ^ / 1 — nsimp 1 — 1 — nsm^cu 
we shall have, adding the three resulting equations together, and dividing by 
(365.) 
n—m 
\[X- 2;.- 2 J =^” [«<I>+«X - »n- (Jd?.y 4+Jdx^/i;-jd Vi.)] 
m (1 — n) 
n Jn—m) 
