Studies on the Motion of Viscous Flows— III 



fii 







U„V' 



f-,, = u„ — + — — + ^—- 



21 "0 



3r "^ 3(9 



Then the summation could have been taken from n = onwards after defining 



^10 = ^20 = ^10 = bjo = c^o = C20 = dio = djo = f 10 = f 20 = 0- The reason for 

 not doing so will become clear shortly. 



These groups can be expressed symbolically in terms of the functions 



S, S„, S^, ..., S, as . .. .. 



CO - . . : 



LS.L„S„ + 2] (L^S„+fn) = .. • (1.35) 



where L is a nojilinear differential operator and Lq is formally identical to L. 

 The operators Ln are linear differential operators which are not formally iden- 

 tical because the coefficients differ for different n, but they nevertheless belong 

 to the same class of differential operators because their structures are similar. 

 The remarkable thing about these operators L^^ is that they depend on the knowl- 

 edge of the functions Si, i :^ 1, 2 n- 1. In Eq. (1.27) we noted that the 



operators L^, i = l, ..., n were formally identical. Moreover, they do not 

 depend on the knowledge of the solutions u^ in any way. Thus the linear differ- 

 ential operators L-, i = 1 n being formally identical, and having the 



same structure and not depending on the functions u^,! = 1 n- 1, form a 



subclass of the class of linear differential operators L,^ which have the same 

 structure and which depend in some way on the functions S^, i = 1, . . . , n - l. 

 One may therefore view Eq. (1.35) as a generalization of Eq. (1.27) for a non- 

 linear case. 



To find uj in Eq. (1.27) a rule was prescribed in the form of Eqs. (1.28). 

 The rule essentially states that not only the sum X(LjUi-fi)= (Lu-f) is 

 zero, but that the individual elements of the sum are also zero. Analogously, 

 we now prescribe the same rule for Eq. (1.35). This gives the equations 



L„s„ = ■ . ;. , ' (1.36) 



K^n + f, = , n = 1 . ■' (1.37) 



In the case of Eq. (1.27), it is proven that such a procedure will give uj, the sum 

 of which is the required solution. We have no such proof for (1.37), but the 

 strong analogy intuitively leads us to believe that such may be the case. If we 

 believe that our equations truly represent the physical processes, then the solu- 

 tion to these equations must represent the observed facts. Conversely, the ob- 

 served facts can be regarded as describing the mathematical solution which is 

 actually 'realized.' This gives us a possibility of a posteriori verification of 

 our procedure by comparison of the solutions so obtained with observed facts. 

 Indeed, the proof of the validity of such a procedure is here heuristic in nature. 



537 



