of Elliptic and Trochoidal Arcs, fyc. 23 



vals (the number being the greater the closer the required 

 approximation). Measure the rolling radii corresponding to 

 and <j>, and to each of the intermediate angles, and let them be 

 denoted by r , r v r^, &c. Then the mean rolling radius is 

 approximately for 2n intervals (by Simpson's first rule) 



rm=Z 6n ( r ° + 4ri + 2r<2 + ^ + * * ■ + 2f2n ~ 2 + 4i7%2n ~ 1 + T ^' y 

 and for 3n intervals (by Simpson's second rule), 



r m — g^( r o + 3r i + 3r 2 + 2r 3 + ... + 2r 3re _ 3 + 3?' 3 „_ 1 + Sr 3 „_ 1 + r 3n ); 



and the arc described by the tracing-point is approximately 

 equal to a circular arc of the radius r mi subtending the angle <j). 



Rule I. To construct a circular arc approximately equal to a 

 given arc G J) (Plate I. fig. 1), not exceeding a quadrant, of 

 an ellipse whose semiaxes A and O B are given. 



In fig. 2, draw a straight line, in which take B F = B and 

 FG=OA. Bisect it in H; and about that point, with the 



radius HF=HK= , describe a circle. Mark the points 



c and d in that circle, by laying off Ec = OC, and E^=OD. 



Then divide the arc cd into 2n or 3n equal intervals, as the 

 case may be, and measure the distances from the ends of the arc 

 and the points of division to G : these will be rolling radii of 

 the ellipse, as generated by rolling a circle of the diameter E H 

 inside a circle of the diameter E G, the tracing-point being at 

 the distance H F from the centre of the rolling circle ; and the 

 Simpsonian mean (as it may be called) of those rolling radii will 

 be the radius of the required circular arc. 



Then in fig. 1 describe a circle about with the radius A ; 

 through C and D draw straight lines parallel to B, cutting 

 that circle in V and A; join O T, A; and about the centre O, 

 with the mean rolling radius already found, describe the cir- 

 cular arc M N, bounded by the straight lines OF, O A ; this 

 will be the required circular arc approximately equal to the elliptic 

 arc C D. 



The same circular arc may be drawn, if required, in fig. 2 as 

 follows : — From any convenient point L in the circumference of 

 the circle F K, draw straight lines through c and d; and about 

 L as a centre, with the mean rolling radius already found, de- 

 scribe the circular arc mn bounded by those lines. 



Rule II. To construct a circular arc approximately equal to 

 a given trochoidal arc, not exceeding the arc between a crest of the 

 trochoid and the adjoining hollow. 



In fig. 2 make G H= the radius of the rolling circle which 



