452 DK. J. G. LEATHEM ON SOME APPLICATIONS OF 



boundary, and if the boundary is of the character shown in fig. 4 the rate of displace- 

 ment is 2lV, where 21 is the breadth of the semi-infinite straight portion on the left. 



Accordingly 



............ (33) 



If the boundary is as in fig. 3, I = and so S = ; it is in any case clear that the 

 motion of a finite rigid internal boundary cannot give rise to a " source-term " in the 

 expansion of w' for z great. 



Thus in order to secure that the transformation (29) shall correspond to a 

 configuration like that of fig. 3, the constants must be arranged so as to make 

 S zero. 



17. The Doublet-Term, and the Coefficient of Inertia. If S is zero the formula (-30) 

 takes the form 



which is equivalent, to the same degree of approximation, to 



dw = (V -Dz- 2 ) dz, 

 and so to 



w^Vz + Dz- 1 ........... (34) 



When by superposition the liquid is brought to " rest at infinity " and the internal 

 boundary is moving to the right witli velocity V, the first approximation to the 

 motion at great distance consists of a " doublet-term," namely, 



w ' = Dz- 1 .......... (35) 



Considering for a moment the translational motion, with velocity V parallel to the 

 axis of x, of a finite solid or " ship " which is not necessarily symmetrical about a line 

 in the direction of motion, it is seen that the full expression, in terms of polar 

 co-ordinates, for the velocity-potential <j> of the surrounding liquid may be written 

 in the form 



0' = r-'(D cos + E sin 0) + Z m r-' n cos(m0-f a,,,) (m = 2, 3, ...). . (36) 



Let GREEN'S- theorem be applied to the functions 0' and x in the region bounded 

 internally by the solid moving boundary, the " ship," and externally by any curve. 

 It then appears that 



where ds is the element of arc, and cm the element of outward drawn normal, has the 

 same value for the internal solid boundary as for any surrounding curve. In 

 particular the surrounding curve may be taken to be a circle whose radius tends to 

 indefinite greatness, and for this it is seen that the terms of type r~ m cos(m0 + m ) 

 make no contribution to the value of the integral, so that the total value is 27rD. 



