334 DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 
Or the loop of a lemniscate is equal to the difference between the circumference of 
Stt 
the spherical parabola whose transverse arc is -^5 ^ semicircle. 
When a quadrant of the spherical parabola is taken, or when the point of contact Q 
coincides with the extremity of the principal minor arc of the curve, we shall have <£=^. 
Since in this case RQ=PQ, FV=F^V, therefore 
j(/,=OFV=OF^V, or RF^V=jt>o+v. As V is the pole of 
RP, and F^ is the pole of AR, the point R is the pole 
of VF,. Hence RF^V is a right angle, bub/.+i'=RF^V, 
whence As tan(<p— when ?>= 2 ’ 
1 
Fig. 8. 
1 T/' • 1 • . / tanju. 
tan[jb=-^ If in the expression tanr=-- ^^ ^ 
sin^ju. 
given in (58.), we substitute this value of tan|M/, we 
TT 
shall get tan7-=l, or 
Hence as two quadrants of a spherical parabola 
are together double of one, we shall have, writing the integral in the 
9 
€ 
d|(x. 
Vi 
VI 
or 
fit _ f ‘ 
Jo VI Vo 
-i/'i \i 
tan 
( 7 )’ 
■✓I' 
( 68 .) 
Now when i is nearly 1, J^-^=log Taking this expression between 
-1 /i\i . . 1 V] 
the limits jm<= 0, and|«/=tan , we shall have, since sinf/;= - ^^-^ , coSj!A=-^^^:Y ^-5 
neglecting J and its powers when added to l,j being very small, 
1 + sina 2 , Ftan (7)“ d/x , / 2 
whence! 
COSjU, 
Therefore (68.) gives 
XXV. To show that 
VI 
(6i'-) 
If dr!/ 
the amplitudes 4 ' and [/j being connected as before, by the equation tan('v//— fi) =7 tan|ti. 
1 4- siny 1 +j 
Since 
tan'4/= 
cot/x— siny tan/x cotjx— ^tan/x 
* “ resultat fort remarquable, dej^ signsJe par Legendre ; raais nous ignorons comment il y est 
parvenu.” — V erhulst, TraiU EUmentaire des Fonctions Elliptiques, p. 158. 
