806 Prof. W. B. Morton on the Discontinuous 



Taking the origin at D we have along DFG 



z = Q \ exp n . (a — u) 3 du=Q' I exp (i%) (a — u) ~ 3 du 



Jd Jd 



(9) 



If the real part of z vanishes at G (when the planes form 

 a right-angled wedge), then 



£ 



cosx • (a-u)-*du = 0. . . . . (10) 



The fl-integrals can be expressed by elliptic functions, 

 but the formulae are clumsy and inconvenient for purposes 

 of numerical calculation. For the ^-integrals one is obliged 

 to have recourse to mechanical or arithmetical quadrature. 

 I have carried out the calculations * for a rectangular wedge 

 whose faces make angles of 45° with the stream. For this 

 purpose the integrals were first transformed so as to cover 

 the range from to ^tt of an angular variable. 



Putting u= —cos 26 in (5) the equation giving/ becomes 



•rr 



-/'.( (-^-cos26 , )-^(-(7~cos2i9)-M<9 



= |+j 2 cos26»(-i-cos2(9)-^-^-cos2(9)-*^. (11) 



(■— d, —g, ate positive quantities greater than unity). 



The two integrands having been tabulated for 2^° intervals 

 the integrals were calculated approximately by Simson's rule 

 and /found. It is to be noticed that the values assumed for 

 d and g must be such that the resulting value of f lies 

 between them. To get a rough idea of the restriction thus 

 imposed on d, g, we may reason as follows. We get an 

 approximation to the integral on the left of (11) by giving 

 the integrand the constant value (dg)~', corresponding to 

 the half-way value of 6 : this gives 7r/2 \l dg. The integral 

 on the right has a small negative value, say — Air. Thus 

 we get as an approximate value 



-/=V^(i + A), 



and this has to lie between — g and — d. It follows that 



* I am much indebted to Miss L. Beck, B.Sc, for lie]p in carrying- 

 out the computations and integrations. 



