<?+$(««) 



c/>0/3) 



W) 



COS a 



cos/3 



cos/3' 



1 



1 



1 



Somograpliy to Discontinuous Motion of a Liquid. 133 



The equation connecting q with the angles can be written 

 in a similar form, viz. : — 



= 0. 



This is equivalent to equation 28, he. cit., where x — ^tt cosec a 

 = — ( Jp -f q cosec a)ir. It can be represented by a nomogram 

 as before, except that now a separate curve must be used for 

 each angle of incidence a. The function $(##), which is 

 plotted against cos 9, has different forms according as is 

 less than, equal to, or greater than a. 

 For 



0< a, it. <f>(*0) = - sin (a + 0) log sin \ [a + 9) — sin (a — 0) log sin J(«— 0) 

 + sin « cos log J sin + -= cos (a — 9). 



a 

 = oi, 7r. <£(aa)= — -J sin 2a log 2 sin a. 



0>a, 7r.<J>(a0) = — sin(a + 0) log sin i (a + 0) + sin (0 — a) log sin i(^- a > 



+ sina cos0 log J sin — — cos (9 — a). 



The nomograms for a=90° and a=60° are shown on 

 figures 3 and 4. There is a sudden decrease, of magnitude 

 unity, in the ordinate when 9 passes the value a ; the point 

 corresponding to a is the middle point of this discontinuity. 

 The required value of q is the distance from this point 

 measured vertically upward to the chord joining the points 

 /3/3'. 



When the angles /3/3' are known, the principle of conser- 

 vation of momentum gives the breadths of the jets, b, (1 — b) y 

 into which the incident stream is split, since 



b cos fi+ (1 — b) cos/3' = cos &. 



The same nomographic method can be used to find the 

 position of the point on the strip at which the stream-line 

 divides, given by its distance r from the centre of the strip 

 (fig. 1), and also the coordinates of points on the free 

 surfaces of the jets. 



The equation for r is identical in form with the equation, 

 for p, the function now being given by 



7r/(<9) = -(? -0^ sin 0-cos 9 log sin 0. 



