304 Mr. W. J. M. Rankine on Plane Water -lines, 



which become sharper in their forms as they are more distant 

 from the primitive water-line of the solid. The only exact 

 water-lines whose forms have hitherto been completely investi- 

 gated, are those generated by the cylinder in two dimensions, 

 and by the sphere in three dimensions. In addition to what is 

 already known of those lines, the author points out that when a 

 cylinder moves through still water, the orbit of each particle of 

 water is one loop of an elastic curve. 



4. The profiles of waves have been used with success in prac- 

 tice as water-lines for ships, first by Mr. Scott Russell (for the 

 explanation of whose system the author refers to the J Transac- 

 tions of the Institution of Naval Architects } for 1860-62), and 

 afterwards by others. As to the frictional resistance of vessels 

 having such lines, the author refers to his own papers — one read 

 to the British Association in 1861 and printed in various engi- 

 neering journals, and another read to the Royal Society in 1862 

 and printed in the Philosophical Transactions. 



5. The author proceeds to investigate and explain the proper- 

 ties of a class of water-lines comprising an endless variety of 

 forms and proportions. In each series of such lines the primi- 

 tive water-line is a particular sort of oval characterized by this 

 property — that the ordinate at any point of the oval is propor- 

 tional to the angle between two lines drawn from that point to 

 two foci. (In Plate VI., LB represents a quadrant of such an 

 oval, being its centre, and A one of the foci ; the other focus 

 is at an equal distance to the other side of the centre.) Ovals 

 of this class differ from ellipses in being considerably fuller at 

 the ends and flatter at the sides. 



6. The length of the oval may bear any proportion to its 

 breadth, from equality (when the oval becomes a circle) to infi- 

 nity. (In the Plate the length O L is to the breadth B nearly 

 as 17: 6.) 



7. Each oval generates an endless series of water-lines, which 

 become sharper in figure as they are further from the oval*. In 

 each of those derived lines, the excess of the ordinate at a given 

 point above a certain minimum value is proportional to the angle 

 between a pair of lines drawn from that point to the two foci. 



8. There is thus an endless series of ovals, each generating an 

 endless series of water-lines; and amongst those figures a con- 

 tinuous or " fair " curve can always be found, combining any 

 proportion of length to breadth from equality to infinity, with 

 any degree of fullness or fineness of entrance, from absolute bluff- 

 ness to a knife-edge. 



* As a convenient and significant name for these water-lines, the terra 

 " Oogenous Neoi'ds " is proposed (from 'Sloyevrjs, generated from an egg, or 



oval). 



