88.] MOTION OF AN ELLIPTIC CYLINDER THROUGH LIQUID. 89 



The equation of the curves by whose motion parallel to x this 

 system of stream-lines could be produced, is by (39) 



y = k(0 l -0J + C (40), 



where k is written for ^ . 



If we put (7=0 we obtain an oval curve symmetrical with respect 

 to x and y. The curve corresponding to any other value of C is 

 symmetrical with respect to the axis of y, and has the line y = G 

 as an asymptote towards x = + oo. The curves for which C is the 

 same in magnitude, but of opposite sign, are symmetrically situated 

 on opposite sides of the axis of x. The points (+ a, 0) which in 

 Example 3 are taken as origins of 1? # 2 , may be called the foci of 

 the above system of curves. By varying the constants k and a 

 the forms of the curves can be varied indefinitely. From their 

 resemblance (within certain limits as to the relative magnitudes 

 of k and a) to the lines of ships, they have been called bifocal 

 neoids, by Prof. Rankine*, who investigated their properties with 

 a view to obtaining theoretical guidance as to what proportions 

 are to be observed in designing a ship in order to reduce as much 

 as possible the resistances due to waves, surface-friction, &c. 



(d) Let f, rj be two new variables connected with x, y by the 

 relation 



x + iy = c sin (f + 117). 

 This gives 



x = c sin f . 



Eliminating f, we have 



c 2 

 and eliminating 77, 



Hence the curves f = const., 77 = const, are confocal hyperbolas and 



* Phil. Trans., 1861, 



