On the Approximate Drawing of Circular Arcs of given Lengths. 285 



indefinite number of circles „. 9 



AD, AD', &c, all in one 



plane, touch each other at 



the point A. Let H G H be 



a curve cutting off equal 



lengths A G, A D, A D', &c. 



from the straight line and 



from all the circles. Take 



A G for the axis of y and A 



for the origin ; let a be the 



common value of the equal 



lengths A G &c, and 6 the 



angle in circular measure 



subtended by any one of the 



circular arcs A D &c. at its centre. Then the coordinates of 



the curve H G H have the following values : — 



x — 7j(l — cos#); y=^sin#. 



6 



e 



(A) 



The radius of curvature of the curve H G H at the point G is the 



x 2 

 limit towards which ^ r approaches indefinitely when 6 di- 

 minishes indefinitely ; and it is easily found by ordinary methods 

 to be fa. Therefore in the neighbourhood of the point G the curve 

 H G H approximates to a circle of the radius G C = f # = f A G. 



4. Method of Calculating Eri^ors. — The errors are calculated 

 as follows : — If A D in fig. 1 were an arc exactly of the required 

 length, the straight line joining C and D would be of the follow- 

 ing length, 



*/{'*(>-$}> 



but its actual length is -j- ; therefore the absolute error is given 

 by the following formula, 



and the proportionate error by the following formul; 



** + 



<r#}> 



a~ 4 V U^ifl 4/ J 

 _3_ // (l- cos fl) 2 



~ 4 V i e* 



__3_ /f2-2cos<9 

 " 4 V I <9 2 



+ 





sin 6 

 



sin 6 



(B) 



