172 ME. C. W. MEEEIFIELD ON THE COMPAEISON OF HYPEEBOLIC AECS. 
If, therefore, we make the above substitutions in this equation, and divide by \/ — 1 . 
we have, making — sin® sin® (p)^, 
— cos®^.tan^, .tan<p 2 -tan<p 3 (8.) 
Moreover, since we also have 
F<Pi+F^2-F(P3=0, 
it is evident that these equations remain true, if we put for 
E^, E^+^.Fip, 
or for H(p, 
k being any constant whatever. 
If we make k~ — sin®^, U®=H®-- sin®^.E^ represents the arc of a hyperbola. 
In fact, if be the equation to a hyperbola, and we make the ordinate 
sin S 
; 5 /= cos®^.tan we have the abscissa — sin®^ sin®(p). Erom these we may 
obtain by differentiation, 
{ 4- — 
J cos> '/(l-sin^e.sin^^!) 
^ 1 — sin^fl sin^(p , T sin^9.<Z<p 
~J cos®ip V'(l — sin®fl,sin®ip) ‘ ^ j ^/(l — sin^fl.sin^(p) ’ 
or U(p = H9— sin®^.F^. 
If we make sin r=sin ^.sin (p, r is the angle which the normal of the M'perbola makes 
with the axis of x. If we change the variable from <p to r, we have 
U =sin® ^.cos® S 
r. . . 
^(sin^fl — sin^ t)‘ 
an equation which bears a remarkable analogy to the arc of the ellipse referred to its 
tangent, 
E,-E=cos®^. f s- 
J (1 — sin^ S . sin® r)- 
It may be worth while to remark, that the angle of the modulus, represents, in the 
ellipse, the eccentricity, while in the hyperbola it represents the angle between the 
asymptote and the ordinate. 
Eor the comparison of hyperbolic arcs, therefore, we have the equation 
Upi-}-U<p2— U(p3= — cos®^.tan(Pi.tanp)2-f3'ii(P3, (9.) 
answering to the equation for elliptic arcs, 
'E<p^ + E(p 2 — Ep) 3 = sin®^ . sin <pi . sin (p 2 • sin p >3 { f - ) 
Eormula (8.) may be derived from the equations (4.), (5.), (6.) in exactly the same way 
that formula (7.) is derived from the equations (1.), (2.), (3.)*. 
* For the details, see Legendbe, ‘Fonctious Elliptiques,’ vol. i. p. 43, or Moseley “ On Definite Inte- 
grals,” Encyclopaedia Metropolitana, ‘ Pure Mathematics,’ vol. ii. p. 497. 
