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Mr. Wi J. Macquorn Rankine on the [Feb. 10, 



continuous closed curves when unifocal are circular, and when bifocal are 

 blunt-ended ovals, in which the length may exceed the breadth in any given 

 proportions. To obtain a unifocal or bifocal neoid resembling a longitu- 

 dinal line of a ship with sharp ends, it is necessary to take a part only of a 

 stream-line, and then there is discontinuity of form and of motion at each 

 of the two ends of that line. 



The author states that the occasion of the investigation described in the 

 present paper was the communication to him by Mr. William Froude of 

 some results of experiments of his on the resistance of model boats, of lengths 

 ranging from three to twelve feet. A summary of those results is printed 

 at the end of a Report to the British Association on the " State of Existing 

 Knowledge of the Qualities of Ships." In each case two models were com- 

 pared together of equal displacement and equal length j the water-line of 

 one was a wave-line with fine sharp ends, that of the other had blunt 

 rounded ends, each joined to the midship body by a slightly hollow neck — 

 a form suggested, Mr. Froude states, by the appearance of water-birds when 

 swimming. At low velocities, the resistance of the sharp-ended boat was 

 the smaller ; at a certain velocity, bearing a definite relation to the length 

 of the model, the resistances became equal, and at higher velocities the 

 round-ended model had a rapidly increasing advantage over the sharp- 

 ended model. 



Hence it appeared to the author to be desirable to investigate the mathe- 

 matical properties of stream-lines resembling the water-lines of Mr. Froude's 

 bird-like models ; and he has found that endless varieties of such forms, 

 all closed curves free from discontinuity of form and of motion, may be ob- 

 tained by using four foci instead of two. They may be called from this 

 property quadrifoca! stream-lines, or, from the idea that suggested such 

 shapes to Mr. Froude, cycnoids\ that is, swan-like lines*. 



Those lines are not to be confounded with the lines of a yacht having at 

 a distance the appearance of a swan, which was designed and built some 

 years ago by Mr. Peacock, for the figure of that vessel is simply oval. 



The paper contains four chapters. The first three are mainly cinemati- 

 cal and geometrical, and relate to the forms of stream-line surfaces in two 

 and in three dimensions, especially those with more than one pair of foci 

 and surfaces of revolution, to the methods of constructing graphically and 

 without calculation, by means of processes first applied to lines of magnetic 

 force by Mr. Clerk Maxwell, the traces of such surfaces, which methods 

 are exemplified by diagrams drawn to scale, and to the motions of the 

 particles of liquid past those surfaces. The fourth chapter is dynamical : 

 it treats of the momentum and of the energy of the disturbance in the 

 liquid, caused by the progressive motion of a solid that is bounded by a 

 ship-shape stream-line surface of any figure whatsoever ; of the ratio borne 

 by the total energy of the disturbance in the liquid to that of the disturb- 

 ing body when that body displaces a mass of liquid equal to its own mass, 



* Kirri'oeif fss 



