22 : Dr. Rankine on the approximate Graphic Measurement 



if only a' = «| ; hence if 7' be the corresponding 



a 



value of 7, the variable sphere passes through the point (a!, 0, </) 

 on the hyperbola, and the envelope is still a cy elide. The cy elide 

 as derived from the foregoing investigation is thus the envelope 

 of a sphere having its centre on the ellipse, and touching a fixed 

 sphere having its centre on the hyperbola. It also appears that 

 there are, having their centres on the hyperbola, an infinite 

 series of spheres each touched by the spheres which have their 

 centre on the ellipse ; if, instead of one of these spheres we take 

 any four of them, this will imply that the centre of the variable 

 sphere is on the ellipse, andit is thus seen that the cyclide as 

 obtained above is identical with the cyclide according to the ori- 

 ginal definition, viz. the envelope of a sphere touching four given 

 spheres. 



Cambridge, December 5, 1864. 



V. On the approximate Graphic Measurement of Elliptic and 

 Trochoidal Arcs, and the Construction of a Circular Arc nearly 

 equal to a given Straight Line. By W. J. Macquorn Ran- 

 kine, C.E., Lh.D., F.R.SS.L. % E.* 



[With a Plate.] 



THE three following rules are very obvious results of the 

 application of Simpson's method of approximate integra- 

 tion to the rectification of curves generated by rolling ; and it is 

 possible that they may have been already published by other 

 authors ; but as I do not know of any such publication, and as 

 the rules are useful and convenient, I beg leave to offer them to 

 the Philosophical Magazine f. 



The general proposition of which the rules are particular cases 

 is the following well-known one. Let a plane disk of any figure 

 roll on a plane base-line of any figure : let there be a tracing- 

 point in the disk, and let r denote the rolling radius, or distance 

 of the tracing-point at any instant from the instantaneous centre, 

 or point of contact of the disk and base-line. Then while the 

 disk rolls through the angle <£, the tracing-point describes an 



arc of the length ! rd<f). To calculate an approximate value of 

 this integral, divide the angle <j> into either 2n or Sn equal inter- 



* Communicated by the Author. 



't A process nearly identical with that of Rule I. is applied by John 

 Bernoulli to the rectification of the whole ellipse, but not to elliptic arcs. 

 (Johannis Bernoullii Opera, vol, i. § 83.) 



