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IX. On the Comparison of Hyperbolic Arcs. By Charles W. Merrifield. 
Communicated hy the Rev. Dr. Booth, F.R.S. 
Eeceived March 3, — Head March 31, 1859. 
Ak application of Jacobi’s second theorem-the imaginary “ 
to a formula which reduces the comparison of the arcs of hyperbolas to he same 
facility as that of elliptic arcs. The transformation is so easy 
had s^me hesitation in publishing it ; but I observe that my result was not noti ed l^y 
Legea-dke, or by Professor Moseley, or in any more recent work which I have seen. 
Some of its applications, too, are worthy of remark. 
I shall use the ordinary notations : — 
A{&, = F(^, 
The functional equation 
I>i+F?)2-F?)3=0 
is satisfied, as is well known, by either of the three trigonometrical equations- 
cos(p 3 =cos?J.cos(p 2 — sin(p,sin<p 2 C(l — sin"^.sin'*?)3) • • • 
cos(p2=cos(p,cos^3+sin<p,sin<p3 •v/(l — sin'^.sin'fpa) . • • 
cos<p, = cos?)2Cos^3+sin?)2sin<p3 -/(I — sin^^.sin^(pi). • • • 
Dividing each of these by cos 9 .. cos cos ^ 3 , and transposing, they become 
secp)3=sec(p,sec(p2+tan(p,tan<p2 A/(l + cos^^.tan"<p3) • • • 
sec <p2=sec<pisec<p3— tamp, tan (p3 C(l+cos^^.tan"<p2) • • 
secipi = sec<p2sec(p3 — tan(p2tanp)3\/(l+cos"Ctan''<p,). • • 
It will be noticed, that we might pass from one set to the other, directly, by substi- 
tuting sec ^ for cos <p, tan ^ for sin <p, and cos ^ for sin These substitutions con- 
stitute Jacobi’s second theorem. They convert 
d(^ . , V ^ .d^ 
(A.) 
( 1 -) 
(2.) 
( 3 .) 
(!•) 
( 5 .) 
( 6 .) 
(1 — sin® 0 . sin^ip)^ ' (1 — sin®0 . sin®<p)^ 
(l-sirfAsin»?.)*# into Asiu*?)! 
and 
Now, calling J(l- sin® sin" E(p, we know that 
E(p, + E<p2 — E<p3= sin" 6 sin cp, . sin <p2 . sin <p3 
2 A 2 
( 7 -) 
