460 Royal Society : — 



colour of rubies and other minerals coloured red by chromic oxide. 

 To others, like the emerald, it imparts a green colour ; and, on the 

 whole, it acts on light in such a variable manner, according to the 

 presence of other substances, that the spectra may be made use of as 

 a means of identifying particular minerals, though they do not pre- 

 sent any thing like such striking anomalies as those met with in the 

 compounds of zirconia with the oxides of uranium. 



" On the Mathematical Theory of Stream-lines, especially those 

 with four Foci and upwards." By William John Macquorn Rankine, 

 C.E., LL.D., F.R.SS. Lond. and Edinb., &c. 



A Stream-line is the line that is traced by a particle in a current 

 of fluid. In a steady current each individual stream-line preserves 

 its figure and position unchanged, and marks the track of a filament 

 or continuous series of particles that follow each other. The mo- 

 tions in different parts of a steady current may be represented to the 

 eye and to the mind by means of a group of stream-lines. 



Stream-lines are important in connexion with naval architecture ; 

 for the curves which the particles of water describe relatively to a 

 ship, in moving past her, are stream-lines ; and if the figure of a 

 ship is such that the particles of water glide smoothly over her skin, 

 that figure is a stream-line surface, being a surface which contains 

 an indefinite number of stream-lines. 



The author in a previous paper proposed to call such stream-lines 

 Neo'ids ; that is, ship-shape lines. 



The author refers to previous investigations relating to stream- 

 lines, and especially to those of Mr. Stokes, in the Cambridge 

 Transactions for 1842 and 1850, on the " Motion of a Liquid past a 

 Solid," and of Dr. Hoppe, on the " Stream-lines generated by a 

 Sphere," in the Quarterly Journal of Mathematics for 1856, and to 

 his own previous papers on " Plane Water-lines in Two Dimensions," 

 in the Philosophical Transactions for 1864, and on " Stream-lines," 

 in the Philosophical Magazine for that year. He states that all the 

 neoid or ship-shape stream-lines whose properties have hitherto been 

 investigated in detail are either unifocal or bifocal ; that is to say, 

 they may be conceived to be generated by the combination of a uni- 

 form progressive motion with another motion consisting in a diver- 

 gence of the particles from a certain point or focus, followed by a 

 convergence either towards the same point or towards a second point. 

 Those which are 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 longitudinal 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 sum- 



