16 



Dr. Rankine on Plane Wat ei' -Lines, [Nov. 26, 



fessor William Thomson (made in 1858, but not yet published), as con- 

 taining the demonstration of the general principles of the flow of a liquid 

 past a solid body. 



3. Every figure of a solid, past which a liquid is capable of flowing 

 smoothly, generates an endless series of 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-hnes whose forms have hitherto been com- 

 pletely investigated, 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 practice as water- 

 lines for ships, first by Mr. Scott Russell (for the explanation of whose 

 system the author refers to the Transactions 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. Viewed as plane water-lines, however, 

 the profiles of waves are not exact, but approximate ; for the " solitary wave 

 of translation," investigated experimentally by Mr. Scott Russell (Reports 

 of the British Association, 1844), and mathematically by Mr. Earnshaw 

 (Camb. Trans. 1845), is strictly applicable to a channel of limited dimen- 

 sions only, and the trochoidal form belongs properly to an endless series of 

 waves, whereas a ship is a solitary body. 



5. The author proceeds to investigate and explain the properties of a 

 class of water4ines comprising an endless variety of forms and proportions. 

 In each series of such lines, the primitive water4ine is a particular sort of 

 oval, characterized by this property, that the ordinate at any point of the 

 oval is proportional to the angle between two lines drawn from that point 

 to two foci. 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 infinity. 



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 Hues 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 continuous or "fair" 

 curve can always be found combining any proportion of length to breadth, 



* As a convenient and significant name for these water-lines, the term " Oogenous 

 Neoids" is proposed (from 'Qoyev?}s, generated from an egg, or oval). 



