30 Proceedings of the Royal Irish Academy. 



17. When the sides of the obstacle are convex to the liquid, d% is negative, 

 and it is clear that if a he small the- ratio of the second term to the first in 

 the coefficient of t- is numerically small. Thus if the point P of departure 

 of the free stream-line be supposed to take positions successively further and 

 further from the vertex of the obstacle, - B\ = i-S, where S begins by being 

 positive but keeps yetting smaller. So limy as S is positive the free stream- 

 line is curving sharply inwards into the obstacle, and so the motion is 

 physically impossible. 



But when a is so chosen that S = 0, S\., is small of the order of t, and 

 the free stream-line osculates the curve of the obstacle. The point of 

 departure of the free stream-line in this case may be called P„. "Without 

 closer consideration of the form of the obstacle behind 1' it is not safe to 

 say thai a free stream-line can actually depart from P„ and be clear of the 

 obstacle ; but, so far as the shape between the vertex and ]\ is concerned, J' . 

 determined by S= 0, may be described as the must forward point of the 

 icle from which a free stream-line can break away. 



If a greatei value be assigned to a than thai corresponding to S = 0, 



- r.illy- S may be expected to have become negative, so that - t\„ is of the 

 order of t- and is negative. This would indicate a free Btream-line breaking 

 away with a sharp concavity to the moving liquid. It may be thought thai 

 this is physically less probable than the smooth departure at /'„; but such an 

 opinion is somewhat speculative. Anyhow the sharp outward turn gives 

 increased chance of clearing the hinder pari of tin- obstacle, so that if 

 departure at /'. were im] — ible it might well he possible from a point v< rj 

 near to /', and behind it. 



The theoretical importance of the point /'. is in any case obvious, and it 

 is to be remembered that the corresponding value of a is determined by the 

 relation 



S - X ♦ ~ [ -^ - 0. 



■ (o - ?r' 







18. It is interesting to note that there is available one rather compre- 

 hensive type of curve-factor into which the parametei a enters in such a \\a\ 

 that the condition S is satisfied. This corresponds to ( '„ of 1), § 40, and 



j be written 



?.« Exp.Jlogj*^ 

 when an arbitrary function. 



