1889.] and the Vena Contractu. 7 



If a = 7r/2 the coefficient of z vanishes, and it will be shewn that 

 the remaining disposable constants can be so chosen as to make 

 the coefficients of z'\ z z , &c. (any desired number of them) also 

 vanish. 



Next suppose that c 1 — 0, then to the third order of approxi- 

 mation 



JL (i? + v 1 ) = 1 + 2c 2 z 2 cos 2<x + &c. 



If a = 7r/4 the coefficient of z 2 vanishes, and if there are con- 

 stants enough, the coefficients of z z , z*, &c. could be made to 

 vanish. And generally if c^c^... c m _ 1 vanish, then to the (m + l)th 

 order of approximation 



_L (y + v *) = i + 2c m z Kl cos ma (5), 



and if a = ir/2m the coefficient of z m vanishes, and the coefficients 

 of z m+1 &c. could be made to vanish. 



We have supposed at first, for simplicity, all the powers of z 

 in (I) to exist, but in every case we shall find that many are 

 wanting. For instance, after the first few terms, only even mul- 

 tiples of y may occur — or again multiples of 6y. Again, even if 

 (1) is complete, yet it will be found that if a — tt/2, (3) will 

 consist only of odd, and (4) of even powers of z. This considera- 

 tion not only simplifies the work, but also enables us with greater 

 ease to make (u 2 + v 2 ) constant to a higher degree of approxi- 

 mation. 



4. We shall now shew how to make the velocities on either 

 side of Oy continuous. We shall take the case to which fig. 2 

 refers, where OB is irj2 and w T here the ultimate value of Cx' is 

 OB. I give this case, not only because it is one of the first I 

 solved, but because I believe it exhibits the method under the 

 greatest disadvantages, and yet it will be found that the approxi- 

 mation obtained is very close. 



We will take for the stream function y}r, and the velocities on 

 the left of Oy, 



^ = — u = a r e x cos y + a 3 e BX cos Sy + Xa 2n e 2nx cos 2ny + A 



dy 



-—- = v = a x e x sin y + a 3 e 3x sin Sy + 2<2 2M e 2 ' !X sin 2ny 



(6), 



S indicates summation for all integral powers of n from 1 to oo . 

 Only two odd multiples of y appear in the above, but it must be 

 distinctly understood (in fact it is the characteristic feature of the 

 present method) that any number of any odd multiples of y could 



