Dr. Rankine on the Mathematical Theory of Stream-lines. 461 



mary 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 compared together of equal 

 displacement and equal length : 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 

 mathematical 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 obtained by using four foci instead of two. 

 They may be called, from this property, quadrifocal stream-lines, or, 

 from the idea that suggested such shapes to Mr. Froude, cycno'ids ; 

 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 

 cinematical 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 pro- 

 cesses first applied to lines of magnetic force by Mr. Clerk Maxwell, 

 the traces of such surfaces, which methods are exemplified by dia- 

 grams 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 disturbing body when that body displaces a mass of liquid equal 

 to its own mass, which ratio ranges in different cases from | to 1 ; 

 of the acceleration and retardation of ships as affected by the dis- 

 turbance in the water, and especially of the use of experiments on 

 the retardation of ships in finding their resistance ; and of the dis- 

 turbances of the pressure which accompany the disturbances of mo- 

 tion in the liquid. Up to this point the dynamical principles arrived 

 at in the fourth chapter are certain and exact, like the geometrical 

 and cinematic principles in the three preceding chapters. The 

 results obtained in the remainder of the fourth chapter are in some 

 respects approximate and conjectural, and are to a gseat extent de- 

 signed to suggest plans for future experiments, and rules for their 

 * KvKvoeidrjs. 



