tomography to Discontinuous Motion of a Liquid. 131 



width of the plane strip, p, the angle of incidence a, and the 

 position of the strip with relation to the stream. This last 

 can conveniently be specified by the perpendicular distance, 

 q, of the middle line of the strip from the middle plane of 

 the undisturbed jet. 



The treatment of the question can be simplified by making 

 a slight modification of the analysis as given in Love's paper. 

 The method consists in representing on the upper half -plane 

 of an auxiliary variable u the regions corresponding to 

 the field of motion on the planes of w = (j> + n/r and 

 X2= — \ogq-\-i6, where (<b yjr) are the velocity-potential and 

 stream-function, and (q 6) are the magnitude of the velocity, 

 taken as unity on the free surfaces, and its inclination to the 

 plane of the strip. The point u = oo is taken to correspond 

 to the point at infinity on the incident jet, and the points 

 w=±l to the edges of the strip. St is then found that 

 w=cosa corresponds to the point on the strip where the 

 stream divides, but the variable u at other points has no 

 simple geometrical meaning. If, however, we make u = co 

 correspond to the point where the stream divides, we shall 

 find that u is cos 6 at all points on the free boundaries of the 

 jets. Taking u= ±1 at the edges of the strip as before, the 

 equation connecting O and u now becomes 



dn/du = 1/-V/V— 1, 



o,=\og{u+{a 2 -iy 2 }. 



On the free surface u*< 1 and Q, = i0, so that u = cos 6. 



The constants which appear in the analysis are the values 

 of u which correspond to the points at infinity on the 

 three jets. These now become the cosines of the angle of 

 incidence a and of the "angles of splash" /3/3' (fig. 1). 



It is evident that the new variable u' is connected with the 

 u of Love's paper by the relation 



w' = (l — au) I (a — u) where <x = cos «. 



The equation giving the breadth p in terms of a/3/3' (equi- 

 valent to equation 23, loc. cit.) can be written in the form 



*+/(«) /(/3) W) 



cos a COS yS COS /3' =0, 



111 



where it f{6) =— ir sin — 2 cos 6 log tan \0. 



This suggests at once a way of finding pairs of angles /3/3 f 

 which can be angles of splash for a stream of unit-breadth 

 incident at angle a on a strip of breadth p. 



