Dr. Rankine on Stream-lines. 



25 



a/T^P^ TCP2 



, and in fig. 3 unity \ and the unit line in fig. 1 



FG 

 is OA, in fig. 2 F G, and in fig. 3 P R. 



The degree of precision of the approximation depends on the 

 smallness of the intervals into which the amplitude is divided — 

 the error diminishing somewhat faster than the fourth power of 

 the interval, as Legendre, in the Appendix already referred to, 

 has shown to be the case for every application of Simpson's for- 

 mula. Thus, in the application of Rule III., the following are 

 examples of the greatest proportionate errors (which are always 

 in excess) : — 



Intervals of amplitude. Errors of Rule III. 



45 about 



30 



22J 



15 



400 

 2 5 00 



8 



" 48000 



The errors of Rules I. and II. become smaller than those of 

 Rule III. as the modulus diminishes — that is, as the ellipse ap- 

 proaches to a circle, and the trochoid to a straight line. The 

 following are examples of the errors of Rule I., in the length of 

 a circular arc equal to an elliptic quadrant, the major semiaxis 

 being unity : — 





Eccentricity. 



True length 



(from 



Legendre's 



Tables). 



Approximate length by Rule I. 



Two intervals 

 of 45°. 



Errors. 



Three inter- 

 vals of 30°. 



Errors. 



0-6 

 0-5 



1-350S 

 1-4184 

 1-4675 



1-3538 

 1-4195 

 1-4681 



•0032 

 •0011 

 •0006 



1-3520 

 1-4186 

 1-4678 



•0014 

 •0002 

 •0003 



Glasgow University, 

 November 19, 1864. 



VI. Supplement to a Paper on Stream-lines*. 

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



[With a Plate.] 

 27. nPHE following is a demonstration of the proposition, 

 -i- stated in an Addendum to the previous paper, that 

 all waves in which molecular rotation is null, begin to break when 

 the two slopes of the crest meet at right angles : — 



The profiles of the layers of a series of waves are converted, 

 as is well known, into undulating stream-lines by supposing 



* In the Philosophical Magazine for October 1864. 



