neighborhood of the wall, making the profile fuller. However, if a roughness element 

 is present, the roughness parameter pu k k/jx would be increased indicating early transi- 

 tion. Such experiments would then clarify the relative importance of the two effects on 

 transition: the velocity profile and the roughness elements. When the roughness effect 

 is dominant, the stability theory might actually give the wrong trend for the effect of 

 heating of the surface. 



In conclusion, I wish to summarize my discussion as follows. When the inherent 

 instability of the boundary layer is the primary cause of transition, the process can be 

 described in the five stages outlined in Section 3. The first stages may be called the 

 instability of the boundary layer, while the last stages represent transition proper. Thus, 

 although existing results from the instability theory do throw much light on the transi- 

 tion problem, some further effort is needed to determine quantitatively the behavior of 

 the oscillations of finite amplitudes for the purpose of understanding the transition 

 mechanism. On the other hand, even for practical purposes, the presence of so many 

 factors influencing the transition phenomenon makes it almost necessary to study its 

 basic mechanism. I hope that further joint theoretical and experimental efforts on this 

 fascinating subject will lead us to an even better understanding of the phenomenon and 

 will offer us a better basis for its practical control. 



REFERENCES 



1. Taylor, G. I., Proc. 5th Int. Congr. Appl. Mech., Cambridge, U.S.A., pp. 294-310 (1938). 



2. Tollmien, W., Nachr. Ges. Wiss. Gottingen, Math. Phys. Klasse, pp. 21-44 (1929). 



3. Taylor, G. I., Phil. Trans. A, 223, 289-343 (1923). 



4. Gortler, H., Z. angew. Math. Mech. 21, 250-2 (1941). 



5. Liepmann, H. W., NACA Rep. ACR 128 (1945). 



6. Heisenberg, W., Ann. Phys. Lpz. (4), 74, 577-627 (1924). 



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Math. phys. Klasse, 1, 47-78 (1935). 



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10. Shen, S. F., J. Aero. Sci. 21, 62-4 (1954). 



11. Lin, C. C, Quart. Appl. Math. 3, 117-42, 218-34, 277-301 (1945). 



12. Lin, C. C., Proc. Nat. Acad. Sci., Wash., 40, 741-7 (1954). 



13. Tollmien, W., Nachr. Ges. Wiss. Gottingen, Math.-phys. Klasse, 50, 79-114 (1935). 



14. Meksyn, D. and Stuart, J. T, Proc. Roy. Soc. A 208, 517-26, (1951). 



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sels), 1956. 



16. Grohne, D., Z. angew. Math. Mech. 35, 344-57 (1954). 



17. Landau, L., C. R. Acad. Sci. URSS 44, 139-41 (1944). 



18. Lin, C. C., Hydrodynamic Stability, Cambridge University Press (1955). 



19. Tollmien, W., Z. angew. Math. Mech. 25/27, 33-50, 70-83 (1947). 



20. Wasow, W., Annals of Mathematics 58, 222-52 (1953). 



21. Klebanoff, P. S., Schubauer, G. B., and Tidstrom, K. D., J. Aero. Sci. 22, 803 (1955). 



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nautics (Brooklyn, New York), 41-74 (1955). 



DISCUSSION 



W. Wasow 



I wish to add some purely mathematical remarks that may help to clarify the 

 relation between Lin's new asymptotic technique and the method I employed in my 

 paper [1] in the Annals of Mathematics, v.58, pp. 223-252 (1953). 



Most work on the asymptotic behavior of the solutions of linear differential 

 equations for large values of a parameter A is based on the idea of comparing the given 

 differential equation L g [y. A] = with a related differential equation L. r [y, A] = 



363 



