Vortex Theory for Bodtes Moving tn Water 
Consequently, T has to be determined by (3.9) and the 
complementary condition 
nT =0 o>. (39") 
Equation (3.9) is a vectorial Fredholm equation of the 2nd kind. It 
is singular since T= An, X_ being a constant, is a solution of the 
homogeneous equation 
LD ae a T(M') we 
“ = (n A Dia faa lf oo TRRE™ ALD ABA" ia 40 
For (3.9) to have solutions, it is necessary and sufficient 
that its right side - say B (M) - be orthogonal to H on)> 
[f= ads bokovg (3. 10) 
my 
This requirement is fulfilled because Vi and curl Jp are divergen- 
celess inside Dj. Hence, if T' isa particular solution of the com- 
plete equation (3.9), the general solution is 
die — ead Uh Dhak Same ea 
But nT' = Gonist.i°= amt) ion: Ye , and therefore 
PPS) Bion a (3891) 
f : : : 1 
is the only solution which fulfils both (3.9) and (3. 9') (1) 
(1) The proofthat (3.10) is sufficient and that nT’ = const. on 2» 
has been omitted because it is possible to substitute for (3.9) a scalar 
equation which does give rise to no difficulty (see Ch. VI, eq. (6. 6'). 
An equation similar to (3.9) has been considered by J. Delsarte [5] in 
the case of a fluid motion inside a closed vessel. In Ch. V, we will 
deal with an equation (5.7) analogous to (3.9) and (6. 6'). 
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