336 DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 
Adopting for the moment the ordinary notation of elliptic integrals, 
1 — i 2 7 
m=-—c=T-:^-, whence 14-c=:ri^- 
1+7 1+7 
Introducing this notation, the last formula will become 
2n.(-c,^)=F.(4) + r^tan[+i+J^] (79.) 
In the Traite des Fonctions Elliptiques, tom. i. p. 68, we meet with the formula 
n,(», +)+n,(^ ^) = F.(+)+^ tan (80.) 
Now when n= — c, this formula becomes 
2n,(-o,^) = F,(+)+d:^tan[+^^], (81.) 
whence (79.) and (80.) are identical. 
XXVII. Let us now proceed to rectify the spherical parabola by the formula for 
rectification given in (47.), the centre being the pole. For this purpose, resuming 
the formula for rectification established in (41.), and deducing the values of the para- 
meter, modulus and coefficients in that expression from the given relations, 
^ , l-fsiny 1+7 ^ 2 siny 
tama=, — +, tana = 7 ^ =7 — • 
1— siny 1— 7 1— smy 1— 7 
we get 
1 —; 
The parameter, tan^£=Y^ 
1-7 
(82.) 
The modulus, sin;; 
The coefficient 
and etang= 
“1+7 
coS|8 2 
sin« cos« 1 +7’ 
1-7 
1+7 
the coefficient 
cos« cos/3 1 —7' 
sin« 1 +7 
.. ^ 
(S3.) 
Making these substitutions in (41.), the resulting equation will become 
2 f d4/ 
(l-7)r 
-7 
'rj 
-if 
tan 
sin\I/ cos\}/ 
— 1 sin^t}/ 
+7^ 
But from (58.), the focus being the pole, we derive 
C dw, , -'r /tana 
-'J V 1 — ^ sin^^a L V 1 sm 
In (74.) we showed that 
d/A 1C dvj/ 
(84.) 
] 
( 85 .) 
J 
sin®. 
