with Logarithmic Singularities and Free Boundaries. 193 



6. Effective inertia of a ship in a canal. — A compact 

 formula for the longitudinal inertia-coefficient of the ship 

 (assumed to have the same mass as the liquid which it 

 displaces) can be deduced. If X is the impulse required to 

 set up a motion of the ship with velocity V in still water of 

 density p, it can be shown that 



p~ i X = $±\ (4>-x'b<l>l'da:)dt/ 9 .... (6) 



the summation referring to two straight lines of integration 

 across the canal, one (with sign +) ahead of the ship, the 

 other (with sign — ) astern. 



To right and to left w can be expanded in the forms 



Yz -f c + Xc n exp( — mrzr 1 ), Yz + c</ + %c n ' exp {nirzl' l ) , (7) 



where the n's are integers. Only the terms c and c ' con- 

 tribute to the limit of the right-hand side of (6) when the 

 transverse lines tend to infinite remoteness. Hence 



p-^X = 2l(c -c ')=-2lC C "™ \dw-Vdz), 



(8) 



wherein the subject of integration is supposed to be expressed 

 in terms of f by means of formulae (3). 



7. Electncal flow in a flat conducting strip pierced by a 

 symmetrical hole. — The analysis of the previous article may 

 be interpreted in terms of two-dimensional electric flow, 

 (f> being taken to be the product of the electric potential and 

 the constant specific conductivity, and the " ship " being 

 represented by a hole cut in the strip. 



In this state of flow let dz x be the element of geometrical 

 displacement corresponding to d<j> when ^ = ; and let dz 2 

 be the corresponding element in what would be the flow if 

 there were no hole in the strip, in which case dw would 

 equal Ydz. Then 



and the latter integral represents the limit of the difference 

 between the lengths of ihe strip, with and without the hole, 

 which correspond to the same difference of potential. In 

 fact it represents the increase of the ohmic resistance of the 

 strip, due to the presence of the hole, in terms of the 

 resistance per unit length. 



Phil. Mag. S. 6. Vol. 31. No. 183. March 1916. P 



