286 Dr. Rankin e on the Properties 



is determined by the ratio which its radius of curvature bears to 

 the unit of measure R, viz. 



v\(u-ir+v*\ mi) 



IT --i+ y -r ■ ■ ^ ll -> 



The waves whose motion is investigated by Professor Stokes 

 in the Cambridge Transactions are of a character intermediate 

 between trochoidal waves and those here considered. 



15. As waves are frequently observed whose figures present a 

 general likeness to that now described, it is probable that a pres- 

 sure approximating to the law expressed by equation (VI.) may 

 be exerted upon them by the wind. 



16. It is evident that a pressure varying according to that law, 

 or nearly so, will be exerted by the bottom of a ship upon the 

 water, when the figures of the buttock-lines, or vertical longitu- 

 dinal sections of her after-body, are exponential stream -lines, or 

 trochoidal waves approximating to them, as in Mr. Scott RusselPs 

 system of shipbuilding. 



Part II. — On Lissoneoids in three Dimensions. 



17. The second part of the investigation relates to the mathe- 

 matical properties of stream-lines of smoothest gliding in three 

 dimensions. The properties of such lines in two dimensions 

 were investigated, and the name " Lissoneoids " proposed for 

 them, in the previous paper already referred to. Their essential 

 mechanical properties are, to have fewer and less abrupt maxima 

 and minima of the speed of gliding of the particles on them than 

 on other stream-lines belonging to the same mathematical class, 

 and to be the fullest lines of their class consistently with not 

 raising more waves than are unavoidable, when they are employed 

 as the lines of a ship. 



18. The mathematical condition which such a stream-line ful- 

 fils is, that at the midship-section or broadest part of the solid 

 to which the line belongs, two points of maximum and one of 

 minimum speed of gliding coalesce into one point. 



19. The investigation shows that the before-mentioned con- 

 dition is expressed mathematically as follows, for any stream-line 

 which at its greatest breadth is parallel to the axis of x. Let u 

 be the longitudinal component, and v and w the transverse com- 

 ponents of the speed of gliding of a particle along the stream- 

 line ; then at the point where that line crosses the midship sec- 

 tion, supposing that we have v = 0, w = 0, the following equation 

 must be fulfilled : 



