of certain Stream-Lines. 287 



The corresponding equation in two dimensions is formed by 

 omitting the term in -=-■ 



Part III. — On some Stream-Lines of Revolution. 



20. The third part of the investigation relates to the stream- 

 lines in which particles now past certain totally immersed oval 

 solids of revolution, bearing the same relation to a sphere that 

 the oval neo'ids described in the previous paper bear to a circle. 

 These lie upon a series of surfaces of revolution, and are the 

 sections of those surfaces by planes passing through the axis. 



21. Let the axis of figure be that of x, and let there be two 

 points in it, called foci, situated at the distances -\-a and — a 

 from the origin. The distance a may be called the eccentricity* 

 Let the perpendicular distance of any particle from the axis be 

 denoted by y ; let / be a constant length, called the parameter ; 

 and let b be the radius of a cylinder which is an asymptote to a 

 given stream-line surface. Then the equation of that surface is 

 as follows : 



Or, in another form, let 6 and 6' be the angles which two lines 

 drawn from the given particle to the foci ( + «) and (— a) respect- 

 ively make with the axis of -f x ; then 



b* = y* -p (cos & -cos 0) (X.) 



22. For the primitive oval solid, b = 0; and by giving b 9 a 

 series of values increasing in arithmetical progression, a series of 

 stream-line surfaces are formed, of gradually increasing width, 

 which divide the liquid mass into a series of concentric tubular 

 streams of equal discharge. 



23. The graphic construction of the stream-lines is as follows. 

 From each of the foci draw a set of diverging straight lines, 

 making angles with the axis whose cosines are in arithmetical 

 progression, the common difference being a sufficiently small 

 fraction. Through the network formed by these lines trace dia- 

 gonally a series of curves traversing the two foci. (The equa- 

 tion of each of those curves is cos 6'— cos 6=m, and they are 

 identical with the lines of force of a magnet having its poles at 

 the foci.) Multiply the parameter (/) by the square roots of 

 the terms of the arithmetical progression, and draw a series of 

 straight lines parallel to the axis and at the distances from it so 

 found ; these will be the asymptotes expressed by the equation 

 b =f Vm. Then through the network formed by those parallel 

 straight lines and the before-mentioned series of curves trace 



