824 PROCEEDINGS OF THE AMERICAN ACADEMY 



The integration of this function, of course, involves elliptic functions 

 For the circle a = b, and 



— Sq = I b = b(x — Xq). 



8°. The equation of the hyperbola may be written, x being a varia- 

 ble analogous to the eccentric angle, 



Q = aChx-\-^ Sh X. (15) 



By differentiation 



C'=aShx-\-^ Ch X, 



Tc>' = i^cr Sli-^ X -j- b' Ch^ X 



= V^(a- 4- b-) Hh:' X -\- U\ 

 Hence the expression for the length of the arc is 



X 



« — «o = I y/(a2 _|_ ft-2j vjli^ .^ _|_ y^, (16) 



In the equilateral hyperbola a =z b, and 



« — «o = / a v/2 Sh- X + 1 = a / y/Ch 2x. 



The equation of the hjperbola referred to its asymptotes is 



Q = xa-{-x-^^. (17) 



Whence by differentiation 



q' = a — x—^^. 

 Let Ta=:k = T/3. Then 



T(/ = k^l — 2ar-2SU«p' + ar-4, 

 and 



In the equilateral hyperbola, cos „ = 0, and 



