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L. On the Approximate Rectification of Circular Arcs ; being a 

 second Supplement to a Paper read to the British Association. 

 By W. J.Macquorn Rankine,C..E., LL.D., F.R.SS.L.fyE* 



1. T>ULE. — To rectify approximately the circular arc AB, 



whose centre is at C. 

 Bisect the arc A B in D and 

 the arc AD in E; so that 

 AE = JAB. Draw the 

 straight tangent A F and the 

 radius C E, and produce that 

 radius till it cuts the tangent 

 in F. Draw the straight line 

 FB. Then AF + FB will be 

 approximately equal in length 

 to the arc A B. 



2. Error. — The error is in 

 excess; it isapproximatelyone 



four- thousandth part of the arc, for an arc equal in length to its 

 own radius ; and its value, in fractions of the length of the arc, 

 varies nearly as the fourth power of the angle subtended by 

 the arc. 



3. Demonstration. — Take the radius C A as the unit of length; 

 and let 6 be the angle, in circular measure, subtended by the 



arc A B. Then A F= tan^ ; and it is easily found by trigono- 

 metry that 



FB 



consequently 



V{ 



tan 2 ^ + 8shr 



&' 



AF + FB = tan | -Tl + \/(l + 8 cos 2 ^ \ . 



For brevity's sake, let tan ~ be denoted by t. Then, by de- 

 veloping the preceding expression in powers of /, we obtain the 

 following series, 



3+27 &C -J 



AF + FB 



= 4t( 



but by a well-known development we have 

 AB = = 4*-Tl--| 



2 /4 



+ IT &L\ 



}> 



* Communicated by the Author. 



