192 Lord Rayleigh : Hydrodynamical Notes. 



Equation (4) may be adapted to our purpose by taking 



m = ?i7r/a, (5) 



where n is an integer. Conditions (2) then give 

 A + C = 0, A + Ccos2*-Dsm2*=0, 



!=b+(5:_2)d=o, 



^B + (~ -2)C sin 2* + (^ -2) D cos 2«=0. 



When we substitute in the second and fourth of these 

 equations the values of A and B ; derived from the first and 

 third, there results 



0(1— cos2«)+Dsin2*==0, 

 Csin2a-D(l-cos2a) = 0; 



and these can only be harmonized when cos 2a=l, or a = s7r, 

 where s is an integer. In physical problems, a is thus limited 

 to the values w and 2*7r. To these cases (1) is applicable with 

 C and D arbitrary, provided that we make 



A + C = 0, B+(l-^)D = 0. . . (5) 



Tims 



^ = C^{cos(^-2^)-cos^} 



making 



V^=4g-l)^ 3+ " /s {Coos(^-2fl) + D S i n (^-2^)}. 



When 5=1, a = 7r, the corner disappears and we have 

 simply a straight boundary (fig. 1). In this case n=\ gives 

 a nugatory result. When n=2 3 we have 



S= 1 

 f=Cr 2 (l-cos26)-^2Cy 2 , .... (8) 

 and V 2 ^=4C. When « = 3.. 



