1863.] 



Dr. Rankine on Plane Water -Lines, 



17 



from equality to infinity, with any degree of fullness or fineness of entrance, 

 from absolute bluffness to a knife-edge. 



9. The lines thus obtained present striking likenesses to those at which 

 naval architects have arrived through practical experience ; and every suc- 

 cessful model in existing vessels can be closely imitated by means of them. 



10. Any series of water-lines, including the primitive oval, are easily and 

 quickly constructed with the ruler and compasses. 



1 1 . The author shows how to construct two algebraic curves traversing 

 certain important points in the water-lines, which are exactly similar for 

 all water-lines of this class. One is a rectangular hyperbola, having its 

 vertex at the end of the oval. It traverses all the points at which the 

 motion of the particles, in still water, is at right angles to the water-lines. 

 The other is a curve of the fourth order, having two branches, one of which 

 traverses a series of points, at each of which the velocity of gliding of the 

 particles of water along the water-line is less than at any other point on 

 the same water-line ; while the other branch traverses a series of points, at 

 each of which the velocity of gliding is greater than at any other point on 

 the same water-line. 



12. A certain point in the second branch of that curve divides each series 

 of water-lines into two classes, — those which lie within that point having 

 three points of minimum and two of maximum velocity of gliding, while 

 every water-line which passes through or beyond the same point has only 

 two points of minimum and one of maximum velocity of gliding. Hence 

 the latter class of lines cause less commotion in the water than the former. 



13. On the water-line which traverses the point of division itself, the 

 velocity of gliding changes more gradually than on any other water-line 

 having the same proportion of length to breadth. Water-lines possessing 

 this character can be constructed with any proportion of length to breadth, 

 from V 3 (which gives an oval) to infinity. The finer of those lines are 

 found to be nearly approximated to by wave-lines, but are less hollow at 

 the bow than wave-lines are. 



14. The author shows how horizontal water-lines at the bow, drawn 

 according to this system, may be combined with vertical plane lines of mo- 

 tion for the water at the stern, if desired by the naval architect. 



15. In this, as in every system of water-lines, a certain relation (according 

 to a principle first pointed out by Mr. Scott Russell) must be preserved 

 between the form and dimensions of the bow and the maximum speed of 

 the ship, in order that the appreciable resistance may be wholly frictional 

 and proportional to the square of the velocity (as the experimental re- 

 searches of Mr. J. R. Napier and the author have shown it to be in well- 

 formed ships), and may not be augmented by terms increasing as the fourth 

 and higher powers of the velocity, through the action of vertical disturbances 

 of the water. 



VOL. XIII. 



C 



