Optimum Shapes of Bodtes tn Free Surface Flows 
and £6 . No general methods have been developed for the exact so- 
lution of nonlinear singular integral equations. Further, it may not 
always be possible to satisfy the condition IT ( + 1) = 0, which are 
required on physical grounds, With these difficulties in mind, we pro- 
ceed to discuss some approximate methods of solution. 
IV. LINEARIZED SINGULAR INTEGRAL EQUATION 
The least difficult case of the extremal problems in this 
general class is when the fundamental function F [r j B | is quadratic 
im ol vand 6. thatis 
F(T, 8, § sc.) = ar “;2pTe ees Zr Wage Gs 
in which the coefficients a,b, ... q are known functions of § and 
may depend on the parameters c,, ... Cy. It should be stressed that 
the above quadratic form of F can generally be used as a first ap- 
proximation of an originally nonlinear problem in which F is trans- 
cendental or contains higher order terms than the quadratic. With this 
approximation the integral equation (22) isthenlinearin IT and Bp , 
and reads 
aD be ep Se Be br ce rg) t ei< Ph (26) 
which combines with (9) to provide a set ot two linear integral 
equations, both of the Cauchy type. The necessary condition (24c), 
obtained from the consideration of the second variation, now becomes 
ae jos ee) S- <0 (ss ll eat als) oe (27) 
For the present linear problem (regarding the integral 
equations) two powerful analytical methods become immediately use- 
ful. First, the coupled linear integral equations (9) and (26) can 
always be reduced toa single Fredholm integral equation of the second 
kind. When the coefficients a(t), b( &) and c(é ) of the quadratic 
terms satisfy a certain relationship, the method of singular integral 
equations can be effected to yield an analytical solution in a closed 
form. 
(i) Fredholm integral equation 
RiAI 
