252 Mr. C. Gill on the Rectification of Curves. 



= cosec ft | 26 sin ? -^ (E-H -f T) j . Where E = 



an elliptic arc, the semiaxes of which are 7- and ^- \/b z a* 9 



and its abscissa from the centre = 2 sin $ ; and H = the arc 

 of an hyperbola whose semiaxes are -j%-9 and ~j&- ^ a > 

 and its tangent, T, terminated by a perpendicular upon it 



from the centre = - _________ _ _ . Which between <t> = 0, 



9 99 



V a 9 

 and $ as arc whose sine is = -7- becomes = cosec ft ( 2 a -f 



b* 



(quadrantal arc of the ellipse + difference between the 



infinite arc of the hyperbola and its asymptote) j for the 

 length of the branch M Q B, and (using the under signs) the 



A* 



length of the other branch P Q' Q" = cosec ft ( C 2a + - x 



2 d 



the above factor) hence the whole length of the curve = twice 



2b 

 the sum of these = - - cosec ft (quadrant of the elliptic arc 



+ excess of the asymptote of the hyperbola, infinitely pro- 

 duced). When B is in the circumference, or b = , E be- 

 comes = its abscissa, and T H = semi-transverse of the 

 hyperbola .-. whole length = 8 a cosec ft, the same as Mr. Be- 

 verley's, when ft = 90. 



This integral only applies when B is without the circle, or 



b ~7 a : when b Z > we shall have M'Q = cosec $fdq>( < 2,b cos 9 



a? + b* cos 2 <p \ Q C _ , . b* sin 2 p 



-j -- 3^ = cosec ft < 2 6 sin <f> H -- __ r =^ 



>/*-* sin 2 4>/ I >/a 2 -^sin a p 



b (E H + T) > ; and here the semiaxes of E are -T-, and 



* * i i fl2 cos <P 



, and it, absdssa = -- --- ; the sem.axes 



of H, 1 and ^p?, and T = >1^_- . This, 



a 2 



between <p = and $ = 180, gives the length of half the 

 curve = b cosec ft (semi-periphery of the ellipse + difference 

 between the two asymptotes and the arc of the hyperbola, 



both 



