CONFORMAL TRANSFORMATION TO PROBLEMS IN HYDRODYNAMICS. 453 



At the inner boundary, if I be the x cosine of the outward normal, dx/dn = I, and 

 d<f>'/dn =lV; so the integral is equivalent to 



If A be the area of the section of the ship, 



\lxds = A, 

 and there results the equality 



cfo .......... (37) 



If/) be the density of the liquid, and ra the impulsive pressure required to generate 

 the motion of the liquid instantaneously from rest, nr = p<\> ; consequently 



(38) 



where X is tlie x component of the resultant impulsive pressure exerted by the ship 

 on the surrounding liquid at the instant when the motion was set up. If the mass 

 of the ship M be equal to that of the liquid which it displaces, as would normally be 

 the case, A = /r'M. Hence (37) is equivalent to 



X,,) .......... (39) 



Now if (X, Y) be the impulse required to set up the whole motion, both the motion 

 of the ship and the motion of the liquid, X = MV + X,,; consequently (39) may be 

 written in the form 



D = X/27r/> ............ (40) 



By using y instead of x in the application of GREEN'S theorem it may be shown 

 similarly that 



E = 



Thus the components D, E of tlie doublet represented by the doublet-terms in the 

 expansion of w' or 0' at great distance are proportional to the components of the total 

 impulse which would generate the motion from rest.* 



In the case of a ship which is symmetrical about the line of motion E and Y are 

 zero, and the effective inertia of the system for longitudinal motion is XV" 1 or 

 2 7 r / oV- 1 D. 



* The corresponding theorem in three dimensions was given by the writer in a paper entitled " On 

 Doublet Distributions in Potential Theory," ' Proc. Royal Irish Academy,' vol. XXXII, Sec. A, No. 4, 

 1914, 14. 



