462 



DE. J. G. LEATHEM ON SOME APPLICATIONS OF 



and it gives a configuration of the character indicated in fig. 7, the second and third 

 straight lines being inclined to the first at angles (n-p-q) * and (2n-p-q-p'-g f ) TT 



respectively. 



In order that the three lines should be 



> parallel it would be necessary to have 



n = p + q = 



(72) 



Fig. 7. 

 Hence the transformation 



If it were further desired that the first and last 

 lines should be parts of the same straight line 

 the constants would have to be so adjusted 

 as to make the term in iv~ l vanish in the 

 expansion of dzfdiv for w great. This condition 

 is equivalent to 



which, when combined with (72), gives 



p' = p = n q = n g'. . . (73) 



dz = 



n 7 

 on CLW 



'20 



2"V (ifl 8 - a 8 ) 1 -' (w* - 



(74) 



gives a configuration of the character of the thick line in fig. 8, which, in the hydro- 

 dynamical application, may be duplicated by reflexion. 



Fig. 8. 



The particular case of q = ^ gives a dumb-bell shaped boundary as in fig. 9, 

 and the case of n = q gives a ship with straight sides and pointed ends, as in fig. 10. 



29. A double curve-factor containing a greater number of arbitrary parameters 

 than ^o is 



, . . . (75) 



where X is positive, and a, g, b, c, h, d are real constants in descending order of 

 magnitude. The use of this factor instead of ^ 20 in the applications of the previous 

 article would give greater variety in the possible shapes of boundary. The constants 

 could probably be adjusted so as to make all three straight portions of the boundary 



