1870.] 



On the Mathematical Theory of Stream-lines. 



207 



so as to give a fine red-pink colour by transmitted light, even when so 

 little chromium is present that the glassy bead is scarcely at all green. 

 If too strongly heated the pink tint is lost. This compound is of interest 

 in connexion with the 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 present any- 

 thing like such striking anomalies as those met with in the compounds of 

 zirconia with the oxides of uranium. 



II. u 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. Re- 

 ceived January 1_, 1870. 



(Abstract.) 



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 motions 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 

 Neoids ; that is, ship-shape lines. 



The author refers to previous investigations relating to stream-lines, and 

 especially to these 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 

 uniform progressive motion, with another motion consisting in a divergence 

 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 



