28 1 Dr. Rankine on the Properties 



+ hyp. log m'. Then trace a series of curves diagonally through 

 the intersections of the network already drawn, in such a man- 

 ner as to make m — m'=b for each curve; these will be the 

 required stream-lines. 



TV 



5. The ordinates for which as is an odd multiple of + — are 



asymptotes to all the stream-lines at the negative side of the 

 axis of x, and are also intersected by each stream-line at the 

 point where y = b. 



6. Maximum values of y for all the stream-lines occur on the 

 ordinates where x has the value 0, or any even multiple of +7r. 



7. Minimum values of -\-y and — y occur on each ordinate 

 where x is an odd multiple of + 7r, but for those stream-lines 

 only for which Z>>1. The stream-lines for which Z><1 do not 

 intersect those ordinates. 



8. The stream-line for which b — \ consists of an endless 

 series of equal and similar curves, each adjacent pair of which 

 cut each other at right angles and the axis of x at angles of 45°, 

 in the points where x is an odd multiple of +7r. 



9. Each stream-line for which b<\ consists of an endless 

 series of equal and similar detached curves, having maximum 

 and minimum values of x given by the equation 



-1 + 6 



cos 5?=— e 



10. Each stream-line for which b>\ is made up as follows : — 

 at the positive side of the axis of x } a continuous curve, present- 

 ing an endless series of equal and similar waves; at the negative 

 side, an endless series of equal and similar detached curves. 



11. The wave-line curves thus formed, as they become more 

 remote from the axis of x (that is, as b increases), approximate 

 more and more nearly to the trochoidal form, which is known to 

 be that of free waves in deep water ; and so rapid is that approxi- 

 mation, that though for b = \ the difference between the two 

 kinds of wave- line is very great, it becomes almost undistin- 

 guishable for b = \\. 



12. Quantities proportional to the component velocities of a 

 particle and to the square of its resultant velocity, are derived 

 from the stream-line function as follows : 



db 7 



w= ~r = 1+ e-»cosa? = l +y — b, 



d V I . . (II.) 



db 



dy 



u 2 -f v 2 = 1 + 2e~y cos x -f e~ 2l J 



j . ■ (ni.; 



