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 XLIII. On the Rectification of Curves, By Mr. CHARLES GILL. 



To the Editors of the Philosophical Magazine and Annals. 

 Gentlemen, 



Y/l^HILE investigating the properties of a family of curves, 

 I was lately led to remark, that the elegant property 

 demonstrated by Mr. Beverley, in your Number for June 

 1 826, is capable of considerable extension. Under the idea 

 that anything tending to the simplification of this abstruse 

 problem will be viewed with satisfaction by your mathematical 

 readers, I submit to you the result of my labours. 



The proposition may be more generally enunciated thus : 

 Let C be any point in a given curve line of any order, and B, 

 a point any how given by position : join B C, and draw BQ, 

 C Q to meet in Q, and contain a given angle (/3), C Q being 

 also a tangent to the curve. Then the rectification of the 

 curve which is the locus of the point Q may be generally ex- 

 pressed by f D C cosec /3 . a 6, being the angle B Q makes 

 with a fixed axis, taken at pleasure. 



Let B M be a fixed axis, and let ^ M B C = <p, and 

 C B Q = Q <p. Now whatever be the nature of the curve 

 in which C moves, or the po- 

 sition of the point B, the equa- 

 tion of the curve may be ex- 

 pressed by B C = (<p) a func- 

 tion of the angle M B C, and 

 given lines. Draw B c indefi- 

 nitely near BC, and with cen- M 

 tre B and radius B C describe 

 the small arc Cr; then, in the elementary triangle Crc, we 



(putting d 8 (<p) = d4>.8'(p)),and ACcr = 180-/3-0+p; 

 hence, tan (/3 + 0-<J>) = - -- = - ......... (*) Now 





B Q . W. . sin (/3 + <p), and forming, as before, the ele- 



sin p 

 mentary triangle Qp q, we shall have Qp = BQ.d0 = 



--f .: pq = 



