ME. C. W. M-F1T?1?, T'FTT7,T.T) ON THE COMPAEISON OE HYPEEBOLIC AECS. 173 
In particular, if <p 3 =i^, we have, for the complementary functions, J](p, + '[J(p^=oo , as 
it ought to be, since the whole length of the curve is infinite. 
For duplication, making (p^=ip 2 =co, we have 
U^ 3 -- 2 Ua;=(cos ^l.tan atf tan (p^. 
The same formula serves for bisection, if we obtain from ^3 by the help of the 
elhptic equations. 
Equation ( 8 .) is easily verified at the extremes, making ^=0, 
and we have the known theorem 
tan a+tan/3 — tan (a+/3)=:— tana.tan jS.tan (a+/3). 
If we make 
A^=cos(p, andH9=j3^, 
whence 
(10.) 
J cos iPi ' J cos <P 2 J COS <P 3 ’ 
which is also a particular case of F<Pi+F 92 — F^ 3 = 0 , depending on the particular equa- 
tion 
sec(p3=secipiSec(p2+tan(pitan(p2, (H-) 
which 1 should call the meridioival equation, from its connexion with the common 
formula for meridional parts, and with certain curves on Mercator’s Chart, which I have 
discussed elsewhere. 
I have taken the trouble of deducing ( 8 .) from (4.), (5.), ( 6 .) directly, but the process 
is so exactly parallel to Mr. Moseley’s work, at vol. ii. p. 497 of the work above cited, 
that it would be unnecessary to insert it here. 
A simpler verification may be found as follows : difierentiating with regard to p the 
expression tan p . Ap, we have 
d_ 
d'p 
(tan (p. A^) = 
A(p 
cos^ <p 
sin^ fl sin^ p 
Af 
— f— fA?) — 
cos^(p~ ^ A<p 
v/hence, by integration (no constant needed, since each term vanishes with p), 
tan p. A^=H<p+Ep — Fp (12.) 
If we now add the equations (7.) and (8.) and subtract the equation 
Fp, + Fp2-Fp3=0, 
we should have, substituting (12.), 
tan Pi . Api -f tan p^ . Ap^ — tan p^ . Apa =- sin^ ^ . sin pi . sin p^ . sin Pa — cos* A tan Pi . tan p^ . tan Pg. 
This equation may be easily verified by using the values of Api, Apa, and Apa obtained 
directly from the equations (1.), (2.), (3.), and clearing by means of the quadratic to 
which they all lead, 
l-f2 cos Pi .cos P 2 .COS Pa = cos* Pi + COS* P 2 -f- cos* Pa + sin* 6 . sin* Pi . sin*p 2 . sin* pa . . (1 4- ) 
( 13 .) 
