Superposed Streams of Viscous Liquids. 499 



Thus we have as our final condition 



/>[.(7P(L4-N)/(a + ^B)-f(« + ^B)(^L4-\N + N 1 X)~U'C(L + N) 



- vN{\(\ 2 - P) + (3\ 2 - &*)Nj + 6\N 2 + 6X3} 



+ 2vF{*L+(X+N l )N}] = P '[^P(L' + N0/(« + ^B) 



+ (a + ^B)(-/^L'-VN' + N 1 / N')-^C / (L' + N')--^N / {-V(V 2 -P) 



+ (3\ /2 -P)N/-6\N 2 ' + 6N 3 '} + 2i/'P{^L / + (-V + N 1 / )N'}] . (7) 



Now we can write the equations (4), (5), (6), (7) in the 

 form 



L+N = I/+N', 



2*>L -hn 2 N = 2ytyv'L' + n 2 'N / , 

 ? 1 L+w 3 N = Z/L / +n 3 , N / . 



The eliminant of L^NxL/N' is the required period equation, 



k n x k m! 1 =0. 



2k 2 pv n 2 -2FpV — n 2 ' 



h n 3 — Z/ —n-/ 



11-1 -1 



§7. We can approximate to the solution of this equation 

 on the supposition that v and v 1 are both small. Now it may 

 easily be shown that the term of highest order in n v is of 

 the order v~*, those in l x , n 2 , n 3 are of the order 1. Hence 

 retaining the terms of the two highest orders only in the 

 period equation, it reduces to 



The first approximation is given by 



h-w= 0. 



Now to our order of approximation 



Hence 



O - pf)gP + + p')W - *(Cp - C V>o = 0, 



where a has been written instead of a + t'A'B. 



2 L2 



