114 MOTION OF A LIQUID IN TWO DIMENSIONS. [CHAP. IV 



that experienced by a plane strip of the same area. This ratio is a maximum 

 when a = 100, about, the lamina being then concave on the up-stream side. 

 In the third column the ratio of P to the distance (26 sin a) between the 

 edges of the lamina is compared with %pq Q 2 . For values of a nearly equal 

 to 180, this ratio tends to the value unity, as we should expect, since the 

 fluid within the acute angle is then nearly at rest, and the pressure-excess 

 therefore practically equal to ^pq Q 2 . The last column gives the ratio of the 

 resultant pressure to that experienced by a plane strip of breadth 26 sin a. 



80. One remark, applicable to several of the foregoing 

 investigations, ought not to be omitted here. It will appear at a 

 later stage in our subject that surfaces of discontinuity are, 

 as a rule, highly unstable. This instability may, however, be 

 mitigated by viscosity ; moreover it is possible, as urged by 

 Lord Rayleigh, that in any case it may not seriously affect the 

 character of the motion within some distance of the points on the 

 rigid boundary at which the surfaces in question have their 

 origin. 



Flow in a Curved Stratum. 



81. The theory developed in Arts. 59, 60, may be readily 

 extended to the two-dimensional motion of a curved stratum of 

 liquid, whose thickness is small compared with the radii of 

 curvature. This question has been discussed, from the point of 

 view of electric conduction, by Boltzmann*, Kirchhofff, Topler, 

 and others. 



As in Art. 59, we take a fixed point A, and a variable point P, 

 on the surface defining the form of the stratum, and denote by -^ 

 the flux across any curve AP drawn on this surface. Then ^r is a 

 function of the position of P, and by displacing P in any direction 

 through a small distance Ss, we find that the flux across the 

 element Ss is given by d^/ds . &s. The velocity perpendicular to 

 this element will be ty/hSs, where h is the thickness of the 

 stratum, not assumed as yet to be uniform. 



If, further, the motion be irrotational, we shall have in addition 

 a velocity-potential </>, and the equipotential curves </> = const, will 

 cut the stream-lines -^ = const, at right angles. 



* Wiener Sitzungsberichte, t. lii., p. 214 (1865). 



t Berl Monatsber., July 19, 1875 ; Ges. Abh., p. 56. 



Pogg. Ann., t. clx., p. 375 (1877). 



