328 PROCEEDINGS OF THE AMERICAN ACADEMY 



and the hyperbolic area (doubk'd) is 



A, = ablx,,, — Char„Sha:„.]. (32) 



These areas are those of segments cut by lines parallel to |3. 

 14". In the equation of the parabola 



let Ta = a, T^ :=: b. In this equation of the curve, a is the diameter 

 and |3 is the tangent at the origin. We have 



q' = x« + jS, 

 ^^^^ TVcx>' = TV( -J «p' - x^u^) = TVa^ ^, 



^ = fsinfJx^=:!^[.3_V]; (33) 



this beirm the area of a sector of which the vertex is at the origin. 

 If a ± /3, then 



Transfer the origin to any point on the tangent ^. Let 

 where m is a scalar. The new equation becomes 





Whence 



and 



vt' = xa -\- ^, 

 TVcrnr' = Tvf^ «,3 — (x^ — xm)af\ 



= TVafsixm — "^ )> 



Xo 



a6 sin ° rx-m x^~\'' /n i\ 



