12 Proceedings of the Royal Irish Academy. 



for any motion bounded partly by fixed and partly by free boundaries ; 

 (iii The exact or approximate adaptation of such a general formula to the 

 case of an obstacle of arbitrarily assigned outline; (iii) The choice of such 

 forms of obstacle as shall correspond to liquid motions that can be precisely 

 specified. 



The exact adaptation "f a general formulation to the case of an assigned 

 obsl - bound to depend upon difficult functional equations, upon w 1 



solution further progress must wail : and ii is doubtful whethei any 

 sufficiently practical and general method of approximation has yel been 

 demonstrated. For this one may find some consolation in the reflection 

 that the question is mainly of academic interest, since in practically important 

 applications to problems of design tin- form of the fixed boundary is the 

 quaesitnm, not the datum. 



The primary objective of a comprehensive formulation remains 

 fundamental importance, since such may well W the only possible point 

 of departure for further progress in general theory. The papers in the 

 above li-t present several different types of mathematical formulation, 

 and yet plainly leave room for possible alternativi 



pr nt paper •■!:• eral formulation in terms of conformal curve- 



— functions whose properties the writei hi aed in previous papers. 



lli'- met in terms of a definite integral involving 



a single arbitrary function of a real variable, and it is believed that in 

 this form the pi of the relation between the fundamental variables 



are exhi simply as From this formulation as starting-point 



it i. i feasible to ma] in knowledge, for there 



are obtained formulae which specify t 1 forward points at which free 



stream-lines from an obstacle with smoothly curved si- 



able connexion between the j f these 



1 » > i i j t ~ and the n which thi earn when there 



are no free stream-lines hut the wake" is in rotational motion; and a 

 principle is deduced which may have important bearing on the problem of 

 designing cylinde nee, as, foi example, sinus for aeroplanes. 



:'. The material configuration is repi- n the plane of a complex 



vanable zmx+ iy. T: and the liquid to 



be streaming oast with mit, tor infinite remoteness 



from the the axis of x, 



