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



When 6 becomes a very small angle, the preceding expression is 

 found, by developing sin 6 and cos 6 in powers of 6, to take the 

 following approximate value, 



(C) 



which varies, as already stated, proportionally to the fourth 

 power of the angle subtended by the arc A D. 



5. Use of the Rules. — Amongst the uses to which both the rules 

 given in article 1 can be applied, there may be specified the set- 

 ting out of arcs of given lengths upon the pitch-circles of toothed 

 wheels. Rule I. is applicable also to laying off given distances 

 upon railway -curves. Rule I. may be applied to other curves 

 besides the circle, provided the deviations from uniformity of 

 curvature are not great. The angular extent of the arcs to 

 which the rules are applicable is limited by the degree of accu- 

 racy required in the measurement, regard being had to the re- 

 lations between that angular extent and the extent of error, as 

 stated in articles 2 and 4. 



Glasgow University, 

 August 27, 1867. 



Appendix. 



Rule III. (See fig. 3.)— To 

 rectify approximately the cir- 

 cular arc A B. Draw the chord 

 AB: produce it, and make 

 AC=iAB; about C, with the 

 radius CB, draw acircle; through 

 A draw the straight tangent AD 

 cutting that circle in D ; AD 

 will be approximately equal in 

 length to the circular arc A B. 

 The errors of this rule are ap- \ \ 



proximately equal to those of \ 



the Rules I. and II., but of the c 



contrary sign, the straight line A D being shorter than the are 

 AB. 



