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y. 



NOTE 0J7 CTJEYATURE OE PEDAL AND RECIPROCAL 

 CrRVES. Br BENJAM^ H. STEEDE, M.D. 



[communicated by me. C. J, JOLT, EOYAL ASTEONOMEE OF lEELAND.] 

 [Read June 27, 1898.] 



1. Ie p be tlie radius of curvature at any point of a curve, p' the 

 radius of curvature at point of contact of corre- 

 sponding tangent to reciprocal curve, ^ the angle — j 

 between radius vector from origin and tangent at 

 the point, then 



pp' sin^fj!) = Ic^, 



h being radius of reciprocation, a result (given 

 in Williamson's "Differential Calculus," Miscel- 

 laneous Examples) which follows directly from 

 the equations 



rdr 



P = 



P = —j-y , rp = rjy = tc- 



p p 



- = — , = sm 



d^ dp' 



2, Again, let C be centre of curvature for point P. Draw CM 

 pei-pendicular to radius vector OF, and draw MN perpendicular to 

 the normal FC, Join JSFO, and let this line produced 

 meet the corresponding normal to the reciprocal curve 

 in C 



Then C is centre of curvature of reciprocal curve 

 for point F' which corresponds to point F on original 

 curve. 



Eor, from similar triangles, 



FN.P'C = OF. OP'] 

 or, since 



F]V= PC sin^, and OP = OTcosecff>, 



PC.P'C sin^c^ = OT. OP' = Tc\ 



h being radius of reciprocation. Therefore (by 1), if (7 be centre of 

 one curve, then C will be centre of curvature for corresponding point 

 of reciprocal curve, and vice versa. 



