Mr. W. J. M. Rankine on Plane Water-lines. 305 



9. The lines thus obtained present striking likenesses to those 

 at which naval architects have arrived through practical expe- 

 rience; and every successful model in existing vessels can be 

 closely imitated by means of them, from a Dutch galliot to a 

 racing-boat. 



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

 are easily and quickly constructed with the ruler and compasses 

 as follows. Parallel to the longitudinal axis X, draw a series 

 of straight lines at equal distances apart. Through the foci 

 draw a series of circular arcs A C lf A C 2 , &c. so as to contain a 

 series of angles found by dividing those distances by 



OL 2 -OA 2 

 20A 



Each of those circular arcs indicates the direction of motion in 

 still water of each of the particles that it traverses. Then through 

 the angles of the network formed by the straight lines and cir- 

 cular arcs draw a series of curves ; these will be the required 

 water-lines*. 



The centre of curvature of the oval at L is the focus A. 



11. The following curves, traversing certain important points 

 in the water-lines, are exactly similar for all water-lines of this 

 class, and are easily and quickly constructed with the compasses. 



L M is a hyperbola having a pair of asymptotes crossing the 

 axes at at angles of 45°. It traverses all the points at which 

 the motion of the particles in still water is at right angles to the 

 water-lines. 



L Q N and L P are the two branches of a curve of the fourth 

 order, having a pair of asymptotes which traverse 0, making 

 angles of 30° with OX. A straight line joining L and P makes 

 an angle of 30° with L 0. The two branches cross the axis X 

 at L, making angles of 45°. The branch L Q N 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. The branch L P 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. The axis Y from B to P traverses a series of points of 

 minimum velocity of gliding : from P onwards it traverses a 

 series of points of maximum velocity of gliding. 



13. Every water-line, complete from bow to stern, which passes 

 within the point P, has three points of minimum and two of 

 maximum velocity of gliding; while every water-line which 



* The first employment of a graphic process of this kind is due, it is 

 believed, to Professor Clerk Maxwell, who applied it to certain curves con- 

 nected with electricity and magnetism. 



Phil. Mag. S. 4. Vol. 26. No. 175. Oct. 1863. X 



