24 On the Measurement of Elliptic and Trochoidal Arcs, fyc* 



generates the trochoid ; and describe the small circle F K about 

 H with a radius equal to the distance of the tracing-point from 

 the centre of the rolling circle. Take G7 and GS respectively 

 equal to the perpendicular distances of the two ends of the given 

 trochoidal arc from the base-line on which the rolling circle 

 rolls. Through y and 8 draw straight lines perpendicular to 

 G H, cutting the small circle in c and d. Divide the arc cdvuto 

 2n or 3n equal intervals, and measure the distances from the 

 ends of the arc and the points of division to G ; these will be 

 rolling radii ; and their Simpsonian mean will be the radius of 

 the required circular arc. 



From H draw straight lines through c and d, and about H, 

 with the radius already found, describe the circular arc p, v, 

 bounded by these straight lines : this will be the required circular 

 arc approximately equal to the given trochoidal arc. 



Rule III. To find the radius of a circular arc which shall be 

 approximately equal to a given straight line, and shall subtend a 

 given angle. 



This question is solved by regarding the straight line as form- 

 ing part of an ellipse whose shorter semiaxis is = 0. Let P Q 

 (fig. 3) be the given straight line. Construct the triangle P Q 11, 

 in which Q is a right angle, and P the complement of the 

 given angle; so that R is the given angle itself. Bisect the 

 hypothenuse P R in S, about which point describe a circular 

 arc traversing P and Q. Divide that arc into 2n or Sn equal 

 intervals : measure the distances from its ends and from the 

 points of division to the point R, and take their Simpsonian 

 mean ; this will be the radius R T of the required arc T U, 

 which is approximately equal to the straight line P Q, and sub- 

 tends the angle P R Q. 



Remarks. — It is evident that the processes described in the 

 three preceding rules are graphic methods of approximating to 

 elliptic functions of the second kind, and that, if put in an alge- 

 braical form, they would become identical with the method of 

 approximation to the function E described by Legendre in the 

 Appendix to the first volume of his Traite des Fonctions Ellip- 

 tiques. 



The amplitude of the function is represented by 



<£ = MONoffig. 1, 

 =mLn= — - of fig. 2, 



= QRPoffig3. 

 The modulus in fig. 1 is the eccentricity of the ellipse, in fig. 2 



