618 Lord Rayleigh on the 



effect is thus a continued passage inwards behind A of nega- 

 tive vorticity, which tends to neutralize in this region the 

 original constant vorticity (Z). When the additional vor- 

 ticity at A is negative (fig. 3), the convection behind A acts- 

 in opposition to diffusion, and thus the positive developed 

 near the wall remains closer to it, and is more easily absorbed 

 as A passes on. It is true that in front of A there is a con- 

 vection of positive inwards ; but it would seem that this 

 would lead to a more rapid annulment of A itself ; and that 

 upon the whole the tendency is for the effect of fig. 2 to 

 preponderate. If this be admitted, we may perhaps see in it 

 an explanation of the diminution of vorticity as we recede 

 from a wall observed in certain circumstances. But we are 

 not in a position to decide whether or not a disturbance dies 

 down. By other reasoning (Reynolds, Orr) we know that 

 it will do so if /3 be small enough in relation to the other 

 elements of the problem, viz. the distance between the walls 

 and the kinematic viscosity v. 



A precise formulation of the problem for free infinitesimal 

 disturbances was made by Orr (1907). We suppose that f 

 and v are proportional to e tnt e lpx , where n—p + iq. If 

 V 2 v = S, we have from (6) and (10) 



P={* 2 -2+?0> + W} S > • (18) 



and ;p-^= s < ■"'■ • • • ( 19 > 



with the boundary conditions that v = 0, dvjdy=0 at the 

 walls. Orr easily shows that the period- equation takes the 

 form 



j , S 1 ^rfy.JS 2 «-^flry-j , S 1 «-^^y.j , S 2 ^rfy = 0, . (20) 



where S 1? S 2 are any two independent solutions of (18), and 

 the integrations are extended over the interval between the 

 walls. An equivalent equation was given a little later (1908) 

 independently by Sommerfeid *. 



Stability requires that for no value of k shall any of the 

 q's determined by (20) be negative. In his discussion Orr 

 arrives at the conclusion that this condition is satisfied, 

 though he does not claim that his method is rigorous. 

 Another of Orr's results may be mentioned here. He shows, 

 that p 4- kfiy necessarily changes sign in the interval between 

 the walls. 



* Atti del IV. Congr. intern, dei Math. "Roma, 1909. 



