Superposed Streams oj Viscous Liquids. 497 



The solution suitable for the lower liquid is to a first 

 approximation 



F(y) = L^ + N^==/ (*/), 



where X 2 = k 2 + (« + t£B)/v, 



and the real part of \ is to be taken positive. 

 To a second approximation 



%)=/o«+/i(y), 



where we have 



The particular integral is 



/ 1 (y) = P i /V^ + Qy^, 



where P = ikC . N/4Xv, 



_ _ (\ 2 -P)+4\ 2 ^C 

 y \(\ 2 -F) '4\v 



To a third approximation 



/jW = (% 4 + Sz/ 3 + Ty 2 + Wy)^, 

 where E = »*C P/8\v, &c. 



Thus for the lower liquid we assume 



F(y) = L^ + N^tl + Njy + N^ + N^ + N^*], 

 where K s = (Q + W)/N, 



N, = (P + T)/N, 

 N 3 = S/N, 

 N 4 = ft/N, 



and the stream function becomes 



¥ = By + iQ/ 2 + F(y>^ +a <. 

 For the upper liquid 



■F'(y) = L^-^4N'^^[l + N/ i / + N 2 y + N 3 y + N 4 y], 

 and ¥' = By + JCV + F^>*+-*. 



PM. il%. S. 6. Vol. 20. No. 117. &?p*. 1910. 2 L 



