Vortex Theory for Bodtes Moving tn Water 
Ss. ¥, ' a ! _4 Vv ' ' 
pg(M', t.) pM Hye te ay Yat t) + Vw Oe £ 
] 
(ts oe 
or 
i 
(M!,t_) = -9_ @ (M!,t_) +> a@ r? to INA | ond, 
LO Lk AO sae 
(Fi 10") 
a2 
The unknown function W, has to be determined by the condition that 
ue Woo eee ais: 
Ss Bg (M: : t) reduce to 5 | es (7 bas, 
when case as reduces to case ay 
This implies that W, is harmonic inside D.. 
Since 
u 
j 
the derivative -—~—® (M;,t,)) contains two kinds of terms. The 
terms of the first kind are those due to the variations of the u; 's 
when the rotation of the system S of axes is ignored. Let us denote 
a(t.) &,; (mM) 
the sum of the terms ofthe first kind. The terms of the second kind 
are due to the fact that M; is fixed on ),;, while M! is fixed in 
the space referred to the fixed axes S'. Consequently they sum up to 
-V (M: , t) Vd, (M:, Le 
L237 
