Mr. C. Gill on the Rectification of Curves. 251 



8'(<P)-sin(/3 + 0-<t>) _ 8(4>)cos(|3 + fl-^)]J. But 

 (), S'(<f>)sm(/3+0-<p) = 8($) cos (|3 + 8-<p); hence pgr 

 . sin (|3 + *-?), and Q q = 



. Sn 



cosec /3, and the rectification of the curve =fdQ. B C cosec /3. 



Cor. 1. Hence, when /3 = 90, the expression becomes 

 ^ . B C, the same as Mr. Beverley's ; still however not 

 restricting the pole to be at the vertex of the curves. 



Cor. 2. Hence, while the generating curve and the pole re- 

 main the same, the length ot the tangential curve is inversely 

 as sin /3 ; for B C is independent of |8 ; and d d = d Q + d 

 (/3 -f 6 <p) is also independent of |3. 



Cor. 3. Hence also a neat method of drawing tangents to 

 curves whose ordinates proceed from a fixed point ; for we 

 have only to draw C Q making with B C (the ordinate) an 



1 



C Q is the required 



angle whose tangent = -= 



tangent. 



Example. Let the curve in which C moves be the circle ; 

 and let OC = , OB = b (O being the centre), O B C = <p, &c. 

 then B C = 8 (<t) = b cos 



= - b 

 cos $ K- 



. Hence the curve M' Q 



= / & ($) cosec /3 . dd = + cosec /3 

 cosec /3 1/26 cos <cfp + I d<\ 



= cosec 



